---
abstract: 'The aim of this paper is to study the well-posedness and the existence of global attractors for a family of Cahn-Hilliard equations with a mobility depending on the chemical potential. Such models arise from generalizations of the (classical) Cahn-Hilliard equation due to <span style="font-variant:small-caps;">M. E. Gurtin</span>.'
author:
- 'Maurizio Grasselli[^1], Alain Miranville[^2], Riccarda Rossi[^3], Giulio Schimperna[^4]'
date: 'March 11th, 2010'
title: |
    Analysis of the Cahn-Hilliard equation\
    with a chemical potential dependent mobility[^5]
---

\#1 \#2 \#3 [width \#1pt height \#2pt depth \#3pt]{}

\#1 \#2 \#3 [width \#1pt height \#2pt depth \#3pt]{}

\#1 \#2 \#3 [width \#1pt height \#2pt depth \#3pt]{}

Introduction {#s:1}
============

In this paper, we address the initial and boundary value problem for the following [*generalized Cahn-Hilliard equation*]{}: $$\label{e:1} \chi_t -\Delta \alpha\left(\delta \chi_t -\Delta \chi
+\phi(\chi) \right)=0 \qquad \text{in }\Omega \times (0,T),$$ where $\delta \geq 0$, $\Omega \subset {\mathbb{R}}^3$ is a bounded domain, $T>0$ a finite time horizon, and $\alpha:{\mathbb{R}}\to {\mathbb{R}}$ a strictly increasing function.

The classical Cahn-Hilliard equation reads $$\chi _t-\Delta w=0,\ \ w=-\Delta \chi +\phi (\chi ) \qquad \text{in
}\Omega \times (0,T),$$ where $\chi $ is the order parameter (corresponding to a density of atoms), $w$ is the chemical potential (defined as a variational derivative of the free energy with respect to the order parameter), and $\phi $ is the derivative of a double-well potential. This equation plays an essential role in materials science and describes phase separation processes in binary alloys (see, e.g., [@Cah; @CahH; @NC]).

By considering a mechanical version of the second law of thermodynamics and introducing a new balance law for interactions at a microscopic level, <span style="font-variant:small-caps;">M. E. Gurtin</span> proposed in [@Gu] the following equations: $$\begin{cases}
\displaystyle{
 \chi_t-{\rm  div}(A(\chi ,\nabla \chi ,\chi _t,w)\nabla w)=0,}
 \\
 \displaystyle{
w=\delta (\chi ,\nabla \chi ,\chi _t,w)\chi _t-\Delta \chi +\phi
(\chi )}
\end{cases} \qquad \text{in } \Omega \times (0,T),$$ Taking $\delta $ constant and $A=a(w)I$, with $a: {\mathbb{R}}\to
{\mathbb{R}}$ a positive function, we then obtain an equation of the form , in which $\alpha$ is some primitive of the function $a$.

In the viscous case $\delta >0$, such equations have been studied in [@rossi05; @rossi06]. Therein, results on the well-posedness and the existence of global attractors have been obtained.

Our main aim in this paper is to treat the case $\delta =0$. We also consider the viscous case $\delta >0$ under different (and more general) assumptions on $\alpha $ and $\phi $ from those in [@rossi05; @rossi06]. In particular, we prove the existence of solutions both in the non-viscous case $\delta =0 $ (cf. Theorem \[th:1\]) and in the viscous case $\delta >0$ (see Theorem \[th:2\]). In the latter setting, under more restrictive assumptions on the nonlinearities $\alpha$ and $\phi$, we also obtain (cf. Theorem 3.1) well-posedness and continuous dependence results for (the Cauchy problem for) . For $\delta>0$ we are also able to study the asymptotic behavior of the system and establish the existence of the global attractor (see Theorem \[th:4\]) in a quite general frame of assumptions on $\alpha$ and $\phi$, which may allow for non-uniqueness of solutions. That is why, for this long-time analysis we rely on the notion of generalized semiflows proposed by <span style="font-variant:small-caps;">J.M. Ball</span> in [@Ball97], and on the extension given in [@rossi-segatti-stefanelli08]. Finally, relying on the short-trajectory approach developed in [@malek-prazak], we also conclude the existence of exponential attractors and, thus, of finite-dimensional global attractors. We recall that an exponential attractor is a compact and semi-invariant set which has finite fractal dimension and attracts the trajectories exponentially fast; note that the global attractor may attract the trajectories at a slow (polynomial) rate (see, e.g., [@BabinVishik92; @EFNT; @MZH]).

This paper is organized as follows. In Section 2, we define our notation and give some preliminary results. Then, in Section 3, we state our main results, whose proofs are carried out in the remaining sections. Finally, in Appendix, we introduce the approximation scheme for our problem and justify the a priori estimates (formally) developed throughout the paper.

Preliminaries {#ss:2.1}
=============

#### Notation and functional setup.

Throughout the paper, we consider a bounded domain $\Omega
\subset {\mathbb{R}}^3$, with sufficiently smooth boundary $\partial \Omega$, and write $|\mathcal{O}|$ for the Lebesgue measure of any (measurable) subset $ \mathcal{O} \subset \Omega$. Furthermore, given a Banach space $B$, we denote by $\Vert\cdot\Vert_B$ the norm in $B$ and by $_{B'}\langle\cdot,\cdot\rangle_B$ the duality pairing between $B'$ and $B$. We use the notation $$H:=L^2 (\Omega),\quad
 V:=H^1(\Omega), \quad
 Z:=\left\{ v \in H^{2}(\Omega) \: : \: \partial_{n} v=0 \right\},$$ and identify $ H $ with its dual space $ H' $, so that $
  Z \subset V \subset H \subset V'\subset Z'  $, with dense and compact embeddings. We denote by $ \mathcal{H} $, $ \mathcal{V} $, $ \mathcal{Z}
$, $ \mathcal{V'} $, and $ \mathcal{Z'} $ the subspaces of the elements $v$ of $H$, $ V $, $Z$, $ V' $, and $Z'$, respectively, with zero mean value $ {m}(v) =\frac{1}{|\Omega|}
{ \sideset{_{Z' }}{_{ Z}}  {\mathop{\langle v , 1  \rangle}}} $. We consider the operator $$\label{def:opA}
 A: V \rightarrow V',  \qquad   _{V'}\langle Au, v \rangle _V:=\int_{\Omega} \nabla u \cdot\nabla v  \quad \forall u, v \in V,$$ and note that $ Au \in {\mathcal{V'}}$ for every $ u \in V $. Indeed, the restriction of $ A $ to $ {\mathcal{V}}$ is an isomorphism, so that we can introduce its inverse operator $
    {\mathcal{N}}: {\mathcal{V'}}\rightarrow {\mathcal{V}}$. We recall the relations $$\begin{aligned}
    &&    \label{Aa}
    { \sideset{_{V' }}{_{ V}}  {\mathop{\langle Au , {\mathcal{N}}(v)  \rangle}}} = { \sideset{_{V' }}{_{ V}}  {\mathop{\langle v , u  \rangle}}}\quad \forall u \in V, \; \forall v \in {\mathcal{V'}},
    \\    &&    \label{Bb}    { \sideset{_{V' }}{_{ V}}  {\mathop{\langle u , {\mathcal{N}}(v)  \rangle}}}  =\int_{\Omega} \nabla({\mathcal{N}}(u)) \cdot\nabla({\mathcal{N}}(v)) \,  {\mathrm{d}}x =
{ \sideset{_{V' }}{_{ V}}  {\mathop{\langle v , {\mathcal{N}}(u)   \rangle}}} \quad \forall u, v \in {\mathcal{V'}},\end{aligned}$$ and that, on account of Poincaré’s inequality for zero mean value functions, the following norms on $V$ and $V'$: $$\begin{aligned}
    &&
  \| u {\|_{V}}^{2}:= { \sideset{_{V' }}{_{ V}}  {\mathop{\langle Au , u  \rangle}}}  + \,{m}(u)^2 \quad \forall u \in V,
  \nonumber  \\
 &&
    \| v {\|_{V'}}^{2}:= { \sideset{_{V' }}{_{ V}}  {\mathop{\langle v , {\mathcal{N}}(v -{m}(v))  \rangle}}}  + \,{m}(v)^2\quad \forall v \in V',
\nonumber
    \end{aligned}$$ are equivalent to the standard ones. It follows from the above formulae that $$\| v {\|_{V'}}^2= { \sideset{_{V' }}{_{ V}}  {\mathop{\langle v , {\mathcal{N}}(v)  \rangle}}}=  \|{\mathcal{N}}(v) {\|_{V}}^2
 \quad \forall v \in {\mathcal{V'}}.$$

It is well known that the operator $A$  extends to an operator (which will be denoted by the same symbol) $A: H \to \mathcal{Z'} $. The inverse of the restriction of $A$ to $ \mathcal{H}$ is the extension of ${\mathcal{N}}$ to an operator ${\mathcal{N}}: \mathcal{Z'} \to \mathcal{H}
  $. By means of the latter, we define the space $$\label{e:sob-neg}
\begin{gathered}
 {\mathcal{W}^{-2,q}(\Omega)}:= \left\{v \in \mathcal{Z'}\, : \
{\mathcal{N}}(v) \in L^q (\Omega) \right\} \qquad \text{for a given $q>1$,}
\\
\text{with the norm $\| v \|_{{\mathcal{W}^{-2,q}(\Omega)}}:=\| {\mathcal{N}}(v)\|_{L^q
(\Omega)}$.}
\end{gathered}$$ The following result shows that, for $q\in (2,6)$ (which is the index range relevant to the analysis to be developed in what follows, cf. ), the space ${\mathcal{W}^{-2,q}(\Omega)}$ can be identified with the dual of the space $${\mathcal{W}^{2,q'}(\Omega)}= \left\{ z \in \mathcal{V}\, : \, Az \in
 L^{q'}(\Omega)\right\}\,,$$ $q'$ being the conjugate exponent of $q$. We endow the latter space with the norm $ \|z \|_{{\mathcal{W}^{2,q'}(\Omega)}}:=
\|Az\|_{L^{q'}(\Omega)}$, which is equivalent to the standard $W^{2,q'}$-norm by the (generalized) Poincaré inequality.

\[le:sob-dual\] For $q\in (2,6)$, the operator $\mathrm{J}: {\mathcal{W}^{-2,q}(\Omega)} \to
({\mathcal{W}^{2,q'}(\Omega)})' $ defined by $$\label{def:operator}
{ \sideset{_{({\mathcal{W}^{2,q'}(\Omega)})' }}{_{ {\mathcal{W}^{2,q'}(\Omega)}}}  {\mathop{\langle \mathrm{J}(v) , z  \rangle}}}:=
{ \sideset{_{V' }}{_{ V}}  {\mathop{\langle Az , {\mathcal{N}}(v)  \rangle}}} \qquad \text{for all $z \in
{\mathcal{W}^{2,q'}(\Omega)}$ and $v \in {\mathcal{W}^{-2,q}(\Omega)}$}$$ is an isomorphism.

We preliminarily note that, since $q \in (2,6)$, the conjugate exponent $q'$ belongs to $(6/5,2)$ and, consequently, one has the following embeddings: $$\label{e:embeddings} \mathcal{H} \subset \mathcal{V'} \subset
{\mathcal{W}^{-2,q}(\Omega)}, \qquad L^{q'}(\Omega) \subset \mathcal{V'}, \qquad
\mathcal{Z} \subset {\mathcal{W}^{2,q'}(\Omega)}\,.$$ Clearly, the operator $\mathrm{J}$ is well defined, linear, and continuous, since, for all $z \in {\mathcal{W}^{2,q'}(\Omega)}$ and $v \in
{\mathcal{W}^{-2,q}(\Omega)}$, $$\label{ineq-cont}
\left|{ \sideset{_{({\mathcal{W}^{2,q'}(\Omega)})' }}{_{ {\mathcal{W}^{2,q'}(\Omega)}}}  {\mathop{\langle \mathrm{J}(v) , z  \rangle}}}
\right| \leq  \| Az\|_{L^{q'}(\Omega)} \|{\mathcal{N}}(v) \|_{L^q(\Omega)}
\leq \|z \|_{{\mathcal{W}^{2,q'}(\Omega)}} \| v\|_{{\mathcal{W}^{-2,q}(\Omega)}}\,.$$ Furthermore, for every $v \in {\mathcal{W}^{-2,q}(\Omega)}$, one can choose $z_v =
{\mathcal{N}}(|{\mathcal{N}}(v)|^{q-2}{\mathcal{N}}(v) )$ (note that $z_v$ is well defined and belongs to $\mathcal{V}$, since $|{\mathcal{N}}(v)|^{q-2}{\mathcal{N}}(v) \in
L^{q'}(\Omega) \subset \mathcal{V'}$ by the second of ). Then, $$\begin{aligned}
& { \sideset{_{({\mathcal{W}^{2,q'}(\Omega)})' }}{_{ {\mathcal{W}^{2,q'}(\Omega)}}}  {\mathop{\langle \mathrm{J}(v) , z_v  \rangle}}}=
 \|{\mathcal{N}}(v)\|_{L^q
(\Omega)}^{q},
\\ &
 \|Az_v\|_{L^{q'}(\Omega)}=
\||{\mathcal{N}}(v)|^{q-2}{\mathcal{N}}(v)\|_{L^{q'}(\Omega)}= \|{\mathcal{N}}(v)\|_{L^q
(\Omega)}^{q-1},
\end{aligned}$$ so that $$\|\mathrm{J}(v)\|_{({\mathcal{W}^{2,q'}(\Omega)})'} \geq
\frac{\left|{ \sideset{_{({\mathcal{W}^{2,q'}(\Omega)})' }}{_{ {\mathcal{W}^{2,q'}(\Omega)}}}  {\mathop{\langle \mathrm{J}(v) , z_v  \rangle}}}
\right|}{\|z_v \|_{{\mathcal{W}^{2,q'}(\Omega)}}} = \frac{\|{\mathcal{N}}(v)\|_{L^q
(\Omega)}^{q}}{\|{\mathcal{N}}(v)\|_{L^q (\Omega)}^{q-1}}=\|{\mathcal{N}}(v)\|_{L^q
(\Omega)}= \|v\|_{{\mathcal{W}^{-2,q}(\Omega)}}\,.$$ In view of , we conclude that $\mathrm{J}$ is an isometry. In particular, it is injective and the image $\mathrm{J}({\mathcal{W}^{-2,q}(\Omega)} )$ is closed in $({\mathcal{W}^{2,q'}(\Omega)})'$. To conclude that $\mathrm{J}$ is surjective, we will prove that $$\label{image:dense}
 \text{$\mathrm{J}({\mathcal{W}^{-2,q}(\Omega)} )$  is dense  in
$({\mathcal{W}^{2,q'}(\Omega)})'$.}$$ Indeed, let $\bar{z}  \in {\mathcal{W}^{2,q'}(\Omega)} $ be such that $$\label{e:test-densita}
{ \sideset{_{({\mathcal{W}^{2,q'}(\Omega)})' }}{_{ {\mathcal{W}^{2,q'}(\Omega)}}}  {\mathop{\langle \mathrm{J}(v) , \bar{z}  \rangle}}}=0
\ \ \text{for all $v \in {\mathcal{W}^{-2,q}(\Omega)}$.}$$ In particular, holds for all $v \in
  \mathcal{H}$, so that, also in view of , $$0 = { \sideset{_{V' }}{_{ V}}  {\mathop{\langle A\bar{z} , {\mathcal{N}}(v)  \rangle}}} ={ \sideset{_{V' }}{_{ V}}  {\mathop{\langle v , \bar{z}  \rangle}}}
= \int_{\Omega} \bar{z} v \quad \text{for all  $v \in
\mathcal{H}$\,.}$$ From the above relation, we easily conclude that $\bar{z}=0$, whence .

#### A generalization of Poincaré’s inequality.

The following result will play an important role in the derivation of the *a priori estimates* of Section \[s:3.1\].

\[le-poinca\] Let $X$ and $Y$ be Banach spaces, with $X$ reflexive, and assume that $$\label{e:compact-embed} \text{$X \Subset Y$ with compact
embedding.}$$ Consider $$\begin{aligned}
& \label{funz-g} G: X \to Y \quad \text{a linear,
weakly-weakly continuous functional,}
\\
& \label{funz-psi}
\begin{aligned}
\Psi: X \to [0,+\infty)  \quad & \text{a $1$-positively
homogeneous,} \\
 & \text{sequentially weakly lower-semicontinuous functional.}
 \end{aligned}\end{aligned}$$ Assume that $G$ and $\Psi$ comply with the following [*compatibility condition*]{}: for all $v \in X$, $$\label{e:comp} Gv=0 \quad \text{and} \quad \Psi(v) =0 \ \Rightarrow
\ v=0\,,$$ and that $$\label{e:norm-equivalent} \exists\, C\geq 1 : \ \ \forall\, v \in X
\quad \frac{1}C \left( \| v\|_{Y} + \| Gv\|_{Y} \right) \leq
\|v\|_{X} \leq C\left( \| v\|_{Y} + \| Gv\|_{Y} \right)\,.$$ Then, $$\label{e:poinc-gen} \exists \, K>0: \ \ \forall\, v\in X  \quad
\|v\|_{X} \leq K\left( \| Gv\|_{Y} + \Psi(v) \right)\,.$$

[*Proof.*]{} Assume, by contradiction, that  does not hold: then, there exists a sequence $\{ v_n \} \subset X$ such that, for every $n \in {\mathbb{N}}$, $$\label{e:contradd} \|v_n\|_{X} > n \left( \| Gv_n\|_{Y} + \Psi(v_n)
\right)\,.$$ In particular, this yields that $\|v_n\|_{X} \neq 0$ for all $n$. Letting $w_n:= v_n / \|v_n\|_{X}$ and using the $1$-homogeneity of $\Psi$, we deduce from  that $$\| Gw_n\|_{Y} +
\Psi(w_n) <\frac1n \quad \text{for every $n \in {\mathbb{N}}$}\,,$$ giving $$\label{e:2.1.12}\text{ $Gw_n \to 0$ in $Y$ \ and  \ $\Psi(w_n) \to
0$ as $n \to +\infty$.}$$ On the other hand, by the reflexivity of $X$, there exists a subsequence $\{ w_{n_k}\}$ weakly converging in $X$ to some $\bar{w}$. In view of –, we find $$w_{n_k} \to w \ \ \text{in $Y$}, \qquad Gw_{n_k} {\rightharpoonup}Gw \ \
\text{in $Y$}, \qquad \Psi(w) \leq \lim_{k \to +\infty}
\Psi(w_{n_k})\,.$$ Hence, yields that $Gw=0$ and $\Psi(w)=0$, so that, by , $w=0$. Thus, by and , $$\lim_{k \to +\infty}\|w_{n_k}\|_{X}\leq  C\lim_{k \to +\infty} \left(
\|w_{n_k} \|_{Y} + \| Gw_{n_k}\|_{Y} \right) =0,$$ in contrast with the fact that $\| w_n\|_{X}=1$ for all $n \in {\mathbb{N}}$.

#### A compactness criterion.

Let $$\label{e:setting}
\begin{gathered}
\text{$\mathcal{O} \subset {\mathbb{R}}^d$, $d\geq 1$, be an open set with
$|\mathcal{O}|<+\infty$,}
\\
\text{ $B$ be a separable Banach space, and $ 1 \leq p <+\infty$.}
\end{gathered}$$ We recall that a sequence $ \{ u_n \} \subset L^{p}(\mathcal{O};B)
$ is *$p$-uniformly integrable* (or simply *uniformly integrable* if $p=1$) if $$\label{lour}
  \forall\, \varepsilon > 0 \quad \exists\, \delta > 0:\quad
  \forall\, J
  \subset \mathcal{O} \quad |J| <
  \delta\ \Rightarrow\ \sup_{n \in {\mathbb{N}}} \int_{J}
  \|u_n(y) \|_{B}^{p} {\, \mathrm{d}}y  \leq \varepsilon.$$ We quote the following result (cf. [@Dunford-Schwartz58 Thm. III.6]) which will be extensively used in what follows.

\[t:ds\] In the setting of , given a sequence $
\{ u_n \}
 \subset L^{p}(\mathcal{O};B) $, assume that there exist a subsequence $\{u_{n_k}\}$ and a measurable function $u :\mathcal{O} \to B$ such that $$u_{n_k} (y) \to u(y) \ \  \text{in $B$} \ \ \text{for almost all}\
y \in \mathcal{O}\,.$$ Then, $u_{n_k} \to u$ in $L^{p}(\mathcal{O};B) $ if and only if it is $ p $-uniformly integrable.

Finally, for the reader’s convenience, here below we report the celebrated lower semicontinuity result due to <span style="font-variant:small-caps;">A.D. Ioffe</span> [@ioffe77].

\[th-ioffe\] Let $f: \mathcal{O} \times {\mathbb{R}}^n \times {\mathbb{R}}^m \to [0,+\infty]$, $n,\,m\geq 1$, be a measurable non-negative function such that $$\begin{aligned}
\label{hyp:ioffe1} &
 f(x,\cdot,\cdot) \ \ \ \text{is lower
semicontinuous on ${\mathbb{R}}^n \times {\mathbb{R}}^m$ for every $x \in \mathcal{O}$,}
\\
& f(x,u,\cdot) \ \ \ \text{is convex on  $ {\mathbb{R}}^m$ for every $(x,u)
\in \mathcal{O} \times {\mathbb{R}}^n$.}\end{aligned}$$ Let $(u_k,v_k), \ (u,v): \mathcal{O} \to {\mathbb{R}}^n \times {\mathbb{R}}^m $ be measurable functions such that $$u_k(x) \to u(x)  \quad \text{in measure in $ \mathcal{O}$,} \qquad
v_k {\rightharpoonup}v \quad \text{weakly in $L^1 ( \mathcal{O}; {\mathbb{R}}^m)$}.$$ Then, $$\label{integral-lsc} \liminf_{k \to +\infty}  \int_{\mathcal{O}}
f(x,u_k(x),v_k(x)) \, \mathrm{d}x \geq \int_{\mathcal{O}}
f(x,u(x),v(x)) \, \mathrm{d}x\,.$$

Global attractors for generalized semiflows {#ss:2.1.1}
-------------------------------------------

As mentioned in the introduction, in order to study the long-time behavior of solutions to the generalized Cahn-Hilliard equation  *in the viscous case*, we rely on the theory of *generalized* semiflows introduced by <span style="font-variant:small-caps;">J.M. Ball</span> in [@Ball97]. In order to make this paper as self-contained as possible, in this section we recall the main definitions and results of this theory, closely following [@Ball97].

\[not:phase-space\] The phase space is a (not necessarily complete) metric space $({X},
{\operatorname{d}_X})$, the distance ${\operatorname{d}_X}$ inducing the *Hausdorff semidistance* $\operatorname{e}$ of two non-empty subsets $A, \, B \subset {X}$ by the formula $\operatorname{e}(A,B):= \sup_{a \in A} \inf_{b \in B} {\operatorname{d}_X}(a,b)$.

\[def:generalized-semiflow\] A *generalized semiflow* ${\mathcal{S}}$ on ${X}$ is a family of maps $g:[0,+\infty) \to {X}$ (referred to as “solutions") satisfying the following properties:

(P1)

:   ***(Existence)*** for any $g_0 \in {X}$, there exists at least one $g \in {\mathcal{S}}$ such that $g(0)=g_0$;

(P2)

:   ***(Translates of solutions are solutions)*** for any $g \in
    {\mathcal{S}}$ and $\tau \geq 0$, the map $g^\tau (t):=g(t+\tau),$ $t \in
    [0,+\infty),$ belongs to ${\mathcal{S}}$;

(P3)

:   ***(Concatenation)*** for any $ g $, $ h \in {\mathcal{S}}$ and $\tau
    \geq 0$ with $h(0)=g(\tau)$, then $z \in {\mathcal{S}}$, $ z$ being the map defined by $$\label{def:concaten}
     z(t):= \begin{cases} g(t)  & \text{if $0 \leq t \leq \tau,$}
     \\
     h(t-\tau) & \text{if $ t >\tau$;}
     \end{cases}$$

(P4)

:   ***(Upper-semicontinuity w.r.t. the initial data)*** if $\{g_n \} \subset {\mathcal{S}}$ and $g_n (0) \to g_0, $ then there exist a subsequence $\{g_{n_k}\}$ of $\{g_n \}$ and $g \in {\mathcal{S}}$ such that $
    g(0)=g_0$ and $g_{n_k}(t) \to g(t)$ for all $t \geq 0.$

#### Orbits, $\omega$-limits and attractors.

Given a solution $ g \in {\mathcal{S}}$, we recall that the *$\omega$-limit* $\omega(g)$ of $g$ is defined by $$\omega(g):= \{ x \in {X}\ : \ \exists \{t_n\} \subset [0,+\infty),
\ t_n \to +\infty, \ \text{such that} \ \ g(t_n) \to x \}\,.$$ Similarly, the *$\omega$-limit* of a set $E \subset {X}$ is given by $$\begin{aligned}
\omega(E):=\big\{
 x \in {X}\ : \ &  \exists  \{g_n\} \subset {\mathcal{S}}\
 \text{such that $\{g_n (0)\} \subset E$, $\{g_n (0)\}$ is bounded,
and} \\
 & \exists \{t_n\} \subset [0,+\infty), \ t_n \to +\infty,\
\text{such that $ g_n (t_n) \to x$} \big\}.\nonumber
\end{aligned}$$ Furthermore, we say that $w: {\mathbb{R}}\to {X}$ is a *complete orbit* if, for any $s \in {\mathbb{R}}$, the translate map $w^s$, restricted to the positive half-line $[0,+\infty),$ belongs to $ {\mathcal{S}}$. For every $\,t \geq 0 $, we can introduce the operator $\,{T}(t): 2^{X}\to 2^{X}\,$ by setting $$\label{eq:operat-T}
  {T}(t)E:=\{ g(t) \ : \
g \in {\mathcal{S}}\ \  \text{with} \ \ g(0) \in E\}\quad \text{for all
} E \subset {X},$$ and define, for $\tau \geq 0$, the set $$\gamma^{\tau}(E):= \cup_{t \geq \tau} {T}(t)E\,.$$ The family of operators $\{{T}(t)\}_{t \geq 0}$ defines a *semigroup* on the power set $2^{X}$. Given subsets $U, E \subset {X}$, we say that $U $ *attracts* $E$ if $\operatorname{e}(T(t)E,U) \to 0$ as $t \to
+\infty$. Furthermore, we say that $U $ is *fully invariant* if $T(t)U = U$ for every $t \geq 0$. Finally, a set $\mathcal{A}
\subset {X}$ is the *global attractor* for ${\mathcal{S}}$ iff it is compact, fully invariant under ${\mathcal{S}}$, and attracts all the bounded sets of ${X}$.

#### **Compactness and dissipativity properties.**

Let $ {\mathcal{S}}$ be a generalized semiflow. We say that $
 {\mathcal{S}}$ is

***eventually bounded*** iff, for every bounded set $B \subset
{X}$, there exists $\tau \geq 0$ such that $ \gamma^\tau (B)$ is bounded;

***point dissipative*** iff there exists a bounded set $B_0 \subset {X}$ such that, for any $g \in {\mathcal{S}}$, there exists $\tau \geq 0$ such that $g(t) \in B_0 $ for all $t \geq
\tau$. The set $B_0$ is then called a (pointwise) *absorbing* set;

***compact*** iff, for any sequence $\{g_n \} \subset {\mathcal{S}}$ with $\{g_n (0)\}$ bounded, there exists a subsequence $\{g_{n_k}
\} $ such that $ \{g_{n_k}(t) \} $ is convergent for any $t
>0.$

We note that the notions that we have just introduced are not independent one from another (cf. [@Ball97 Props. 3.1 and 3.2] for more details).

#### **Lyapunov function.**

The notion of a *Lyapunov function* can be introduced starting from the following definitions: we say that a complete orbit $g \in {\mathcal{S}}$ is *stationary* if there exists $x \in {X}$ such that $g(t)=x$ for all $t \in {\mathbb{R}}$ - such an $x$ is then called a *rest point*. Note that the set of rest points of ${\mathcal{S}}$, denoted by $Z({\mathcal{S}})$, is closed in view of **(P4)**. A function $V: {X}\to {\mathbb{R}}$ is said to be a *Lyapunov function* for ${\mathcal{S}}$ if $V$ is continuous, $V(g(t)) \leq V(g(s))$ for all $g \in {\mathcal{S}}$ and $0 \leq s \leq
t$ (i.e., $V$ decreases along all solutions), and, whenever the map $t \mapsto V(g(t))$ is constant for some complete orbit $ g$, then $
g $ is a stationary orbit.

#### **Existence of the global attractor.**

The following theorem subsumes the main results from [@Ball97] (cf. Thms. 3.3, 5.1, and 6.1 therein) and provides the basic criteria for the existence of the global attractor ${\mathcal{A}}$ for a generalized semiflow ${\mathcal{S}}$.

\[thm:ball1\] Let ${\mathcal{S}}$ be an eventually bounded and compact generalized semiflow. Assume that ${\mathcal{S}}$ also admits a Lyapunov function $V$ and that $$\label{rest-bounded} \text{
 the set of its rest points ${Z({\mathcal{S}})}$ is bounded.}$$ Then, ${\mathcal{S}}$ is also point dissipative, and, consequently, it possesses a global attractor. Moreover, the attractor ${\mathcal{A}}$ is unique, it is the maximal compact fully invariant subset of ${X}$, and it can be characterized as $$\label{eqn:attrattore}
{\mathcal{A}}= \bigcup \{\omega(B) \ : \ \text{$B \subset {X}$
bounded}\}=\omega({X}).$$ Finally, for every $g \in {\mathcal{S}}$, $$\label{e:additional} \omega(g) \subset {Z({\mathcal{S}})}.$$

\[rem:restriction\_to\_invariant\_set\] Actually, it is immediate to check that, if ${\mathcal{S}}$ is compact, eventually bounded, and admits a Lyapunov function, then condition  can be replaced by $$\label{e:saab}
\begin{gathered}
 \exists\, \mathcal{D} \subset {X}\,, \ \  \mathcal{D}\neq
 \emptyset\,, \  \ \text{such that}
 \ \ \begin{cases}
 {T}(t) \mathcal{D} \subset \mathcal{D}
\quad \forall t \geq 0,
\\
 \text{the set $ Z({\mathcal{S}}) \cap \mathcal{D} $ is bounded in
${X}$}. \end{cases}
\end{gathered}$$ Then, under these hypotheses, ${\mathcal{S}}$ also possesses a (unique) global attractor ${\mathcal{A}}\subset
\mathcal{D}$ and  holds.

Main results {#s:2}
============

A global existence result for the non-viscous problem {#ss:2.2}
-----------------------------------------------------

#### Assumptions on the nonlinearities.

We assume that $$\label{e:hyp1} \tag{H1}
\begin{gathered}
\alpha: {\mathbb{R}}\to {\mathbb{R}}\qquad \text{is a strictly increasing,
differentiable function such that}
\\
\exists\, p \geq 0, \ \ \exists\, C_1,\, C_2 >0 \, : \quad \forall\,
r \in {\mathbb{R}}\qquad C_1 \left( |r|^{2p}+ 1\right) \leq \alpha'(r) \leq
C_2 \left( |r|^{2p}+1\right)\,.
\end{gathered}$$ Clearly, the latter growth condition entails that $$\label{e:conse1} \exists\, C_3,\, C_4,\, C_5 >0 \, : \quad \forall\,
r \in {\mathbb{R}}\qquad C_3  |r|^{2p+1}-C_4 \leq \alpha(r)\text{\rm sign}(r)
\leq C_5 \left( |r|^{2p+1}+1\right)\,.$$

Concerning the nonlinearity $\phi$, we require that $$\label{e:hyp2}
\tag{H2}
\begin{gathered}
\text{dom}(\phi) = I, \ \ \text{$I$ being an open, possibly unbounded,
interval $(a,b)$, $ -\infty \leq a < 0 < b \leq +\infty$,}
\\
\phi \in {\mathrm{C}}^1 (I),
\\
\lim_{r \searrow a} \phi(r) = -\infty,\qquad \lim_{r \nearrow b}
\phi(r) = +\infty,
\\
\lim_{r \searrow a} \phi'(r) =\lim_{r \nearrow b} \phi'(r) =
+\infty\,.
\end{gathered}$$ We shall denote by $\widehat{\phi} $ (one of) the antiderivative(s) of $\phi$. It follows from the above assumptions that $\widehat{\phi}$ is bounded from below. Hereafter, for the sake of simplicity, we assume that $$\label{phicappucciopos} \widehat{\phi}(r) \geq 0 \quad \text{for all
$r \in I$.}$$ Furthermore, obviously yields that $$\label{e:hyp2-bis}  \exists\, C_{\phi,1} >0 \, : \ \ \forall\, r \in
I \qquad \phi'(r) \geq -C_{\phi,1}\,,$$ namely, $\phi$ is a Lipschitz perturbation of a non-decreasing function. In particular, we will use the fact that there exists a non-decreasing function $\beta: I \to {\mathbb{R}}$ such that $$\label{e:hyp2-aftermath} \phi(r) = \beta(r) -C_{\phi,1} r \qquad
\forall\, r \in I\,.$$ Consequently, $\widehat{\phi}$ is a quadratic perturbation of a convex function. Arguing in the very same way as in [@mirzel04] (where the case $I= (-1,1)$ was considered), it can be proved that, under these conditions, the following crucial estimate holds: $$\label{e:2.2.14} \forall\, m \in (a,b)\ \  \exists\, C_m,\ C_m'>0 \,
: \ \  \forall\, r \in (a-m,b-m) \quad |\phi(r+m)| \leq C_m
\phi(r+m)r +C_m'\,.$$\
Finally, we also assume that $$\label{hyp:3} \tag{H3} \exists\, \sigma \in (0,1), \ \ \exists\,
C_6>0\, : \quad \forall\, r \in (a,b) \quad |\phi(r)|^{\sigma} \leq
C_6 \left( \widehat{\phi}(r) +1\right)\,,$$ and that the following [*compatibility condition*]{} holds between $\sigma$ and the growth index $p$ of $\alpha$ in : $$\label{hyp:4} \tag{H4} \sigma >
\max\left\{\frac{6p-3}{6p+2},0\right\} \,.$$ Hence, if $p\leq 1/2$, then any $\sigma  \in  (0,1)$ is admissible, while if, for instance, $p=1$, then the range of admissible $\sigma$’s is $(3/8, 1)$, and it is $(9/{14},1)$ for $p=2$.

\[not:2.1\] Hereafter, we will use, for every $p\geq 0$, the short-hand notation $$\label{e:index-notation} \rho_p:= \frac{2p+2}{2p+1}, \qquad
\kappa_p:=\frac{6p+6}{2p+1}, \qquad \eta_{p\sigma}=
\frac{6-\sigma}{(3-3\sigma)(2p+1)}\,.$$ For later convenience, we note that $\rho_p$ and $\kappa_p$ are decreasing functions of $p$ and $$\label{e:2.17} 1<\rho_p<2, \qquad 3<\kappa_p <6 \qquad  \text{for
every $p\geq 0$.}$$ Furthermore, it can be checked that $$\label{e:really-necessary} \eta_{p\sigma}>1 \quad \text{for every} \
p \geq 0  \ \text{and for all} \   \sigma >
\max\left\{\frac{6p-3}{6p+2},0\right\}\,.$$

#### The existence result.

We are now able to give the variational formulation of the boundary value problem associated with  in the non-viscous case.

\[p:1\]Find a pair $(\chi,w)$ fulfilling $$\begin{aligned}
& \label{1-var} \chi_t + A (\alpha(w)) =0 \qquad \text{in
${\mathcal{W}^{-2,\kappa_p}(\Omega)}$} \quad {\text{a.e.\ in}}\ (0,T)\,,
\\
& \label{2-var} A\chi + \phi(\chi) =w \qquad {\text{a.e.\ in}}\ \Omega \times
(0,T)\,.\end{aligned}$$

Note that, owing to Lemma \[le:sob-dual\], is equivalent to $$\label{e:difficult}
\begin{aligned}
{ \sideset{_{{\mathcal{W}^{-2,{\kappa_p}}(\Omega)} }}{_{ {\mathcal{W}^{2,{\kappa_p}'}(\Omega)}}}  {\mathop{\langle \chi_t , v  \rangle}}}
 & + { \sideset{_{{\mathcal{W}^{-2,{\kappa_p}}(\Omega)} }}{_{ {\mathcal{W}^{2,{\kappa_p}'}(\Omega)}}}  {\mathop{\langle A
(\alpha(w)) , v  \rangle}}}=0 \\ &   \text{for all $ v \in
{\mathcal{W}^{2,{\kappa_p}'}(\Omega)} \quad {\text{a.e.\ in}}\ (0,T)$.}
\end{aligned}$$

\[th:1\] Under assumptions –, for every initial datum $\chi_0$ satisfying $$\label{hyp:initial-datum} \chi_0 \in V, \qquad
\widehat{\phi}(\chi_0) \in L^1(\Omega)\,,$$ there exists at least a solution $(\chi,w)$ to Problem \[p:1\], with the regularity $$\begin{aligned}
& \label{reg-chi} \chi \in L^2 (0,T;W^{2,6} (\Omega)) \cap L^\infty
(0,T;V), \qquad \chi_t \in L^{\eta_{p\sigma}}(0,T;
{\mathcal{W}^{-2,\kappa_p}(\Omega)})\,,
 \\
& \label{reg-w0} w \in L^2 (0,T;V), \qquad \alpha(w) \in
L^{\eta_{p\sigma}}(0,T;L^{\kappa_p}(\Omega))\,,\end{aligned}$$ fulfilling the initial condition $$\label{e:init-cond} \chi(0)=\chi_0 \quad \text{in $V$.}$$

A formal proof of this result will be developed in Section \[s:3\] and rigorously justified in Appendix.

A global existence result for the viscous problem {#ss:2.3}
-------------------------------------------------

We replace our assumptions – on $\phi$ and its antiderivative $\widehat{\phi}$ by $$\label{hyp:5} \tag{H5}
\begin{gathered}
 \widehat{\phi} : {\mathbb{R}}\to {\mathbb{R}}\quad \text{belongs to ${\mathrm{C}}^2 ({\mathbb{R}})$ and satisfies}
\\
  \exists\, C_7>0\,
: \quad \forall\,r \in  {\mathbb{R}}\quad |\phi(r)| \leq C_7 \left(
\widehat{\phi}(r) +1\right)\,.
\end{gathered}$$ The latter assumption means that we consider potentials with at most an exponential growth at $\infty$, and it clearly yields that $\widehat{\phi} $ is bounded from below. Hence, as in , we again assume that $\widehat{\phi}$ takes non-negative values. Furthermore, as in the non-viscous case we require that $$\label{e:hyp2-bis-visco} \tag{H6}
\begin{gathered}
 \exists\, C_{\phi,2} >0 \, : \ \ \forall\, r \in {\mathbb{R}}\qquad \phi'(r) \geq -C_{\phi,2}\,.
\end{gathered}$$ This and  imply that $$\label{e:lambda-convex}   \ r \in {\mathbb{R}}\mapsto \widehat{\phi}(r)
+\frac{C_{\phi,2}}2 r^2 \ \ \text{is convex and bounded from below.}$$

Let us point out that  yields $$\label{e:phi-add} |\widehat{\phi}(r)| \leq  |\widehat{\phi}(0)| +
|\phi(r)||r|+ \frac{C_{\phi,2}}{2}r^2    \quad \text{for all $r \in
{\mathbb{R}}$.}$$ Indeed, it follows from  and an elementary convexity inequality that, for every $r \in {\mathbb{R}}$, $$\widehat{\phi}(0) - \widehat{\phi}(r) - \frac{C_{\phi,2}}{2}r^2 \geq
-r\left( \phi(r) + C_{\phi,2}r \right)\,,$$ whence we deduce  with straightforward algebraic manipulations.

We will address the analysis of the Cahn-Hilliard equation  in the viscous case under the aforementioned assumptions. The related variational formulation reads

\[p:2\]Given $\delta >0$, find a pair $(\chi,w)$ fulfilling $$\begin{aligned}
& \label{1-var-better}
 \chi_t + A (\alpha(w)) =0 \qquad {\text{a.e.\ in}}\  \Omega \times (0,T)\,,
 \\
 &
 \label{2-var-better} \delta \chi_t+ A\chi + \phi(\chi) =w \qquad {\text{a.e.\ in}}\ \Omega \times
(0,T)\,.
 \end{aligned}$$

#### The existence result.

\[th:2\] Assume , , and . Then, for every initial datum $\chi_0$ complying with , there exists at least a solution $(\chi,w)$ to Problem \[p:2\], with the regularity $$\begin{aligned}
& \label{reg-chi-bis} \chi \in L^2 (0,T;Z) \cap L^\infty (0,T;V)
\cap H^1 (0,T;H)\,,
 \\
& \label{reg-w} w \in L^2 (0,T;V) \cap
L^{2p+2}(0,T;L^\infty(\Omega)), \qquad \alpha(w) \in
L^{\rho_p}(0,T;Z)\,,\end{aligned}$$ and such that $\chi$ satisfies the initial condition .

We refer to Section \[s:3\] for a formal proof of Theorem \[th:2\] and to Appendix for all rigorous calculations.

In addition, we also have the following regularity result, which plays a key role in Section \[ss:2.4\].

\[prop:regularized\] Assume , , and . Assume that, in addition, $\phi$ satisfies $$\label{e:addphi} \widehat{\phi} \in \mathrm{C}^2 ({\mathbb{R}}) \ \
\text{and} \ \ \exists\, C_{\phi,3}>0\,: \ \ \forall\, r \in {\mathbb{R}}\quad |\phi{'}(r)| \leq C_{\phi,3}(1+|r|^4)\,.$$ Then, for all $0 <\tau <T$, the pair $(\chi,w)$ has the further regularity $$\begin{aligned}
\label{e:further-reg-chi} & \chi \in L^\infty(\tau, T; Z) \cap H^1
(\tau,T; V)\,,
\\
& \label{e:further-reg-w}
 \alpha(w) \in L^{\rho_p} (\tau, T; H^3(\Omega))\,.\end{aligned}$$ In particular, if $\chi_0\in Z$, then the above properties hold for any $\tau\in [0,T)$.

\[uniform\] From the proof of Proposition \[prop:regularized\], it is not difficult to recover a uniform estimate of the following form: $$\label{uniform-gronwall} \Vert\chi\Vert_{ L^\infty(\tau, T; Z) \cap
H^1 (\tau,T; V)} +
 \Vert\alpha(w)\Vert_{L^{\rho_p} (\tau, T; H^3(\Omega))}
 \le Q(\tau^{-1},\Vert \chi_0\Vert_V),$$ where $Q$ is a suitable function which is nondecreasing with respect to both arguments.

Well-posedness for the viscous problem
--------------------------------------

#### Continuous dependence on the initial data and uniqueness.

We will prove uniqueness (and continuous dependence) results for Problem \[p:2\] under more restrictive assumptions on $\alpha$ and on the growth of the function $\phi$. In particular, we are going to consider two sets of assumptions.

First, we will suppose that $\phi$ behaves like a polynomial of degree at most $3$. For the sake of simplicity and without loss of generality, we will carry out our analysis in the case when $\phi$ is the derivative of the double-well potential $\widehat{\phi}(r)=(r^2-1)^2/4$. Furthermore, we will replace  by $$\label{e:hyp1-bis} \tag{H7}
\begin{gathered}
\alpha: {\mathbb{R}}\to {\mathbb{R}}\qquad \text{is a strictly increasing and
differentiable function such that}
\\
 \ \exists\, C_{9},\, C_{10} >0 \, : \quad \forall\, r
\in {\mathbb{R}}\qquad C_{9} \leq \alpha'(r) \leq C_{10}\,,
\end{gathered}$$ and – by $$\label{e:hyp7-bis} \tag{H8} \phi(r) =r^3-r \qquad \forall\, r \in
{\mathbb{R}}\,.$$

\[th:3\] Assume  and . Let $\chi_{0}^{1}$ and $\chi_{0}^{2} $ be two initial data for Problem \[p:2\] fulfilling  and set $M_{*}:= \max_{i=1,2} \{\| \chi_{0}^{i}{\|_{V}}\}
    $; let $ \chi_{i} $, $ i=1,2 $, be the corresponding solutions. Then, for every $\delta>0$, there exists a positive constant $ S_{\delta} $, also depending on $$\label{e:only-depe}
    \text{$ M_{*} $,
    $ T $, $ |\Omega| $, $C_{9}$, and $C_{10} $,}$$ such that $$\label{contdepV}
\| \chi_1 (t) - \chi_2 (t) \|_{V}   + \| \chi_1 - \chi_2 \|_{H^1
(0,t;H) \cap L^2 (0,t;Z)}
 \leq
    S_{\delta}   \|\chi_{0}^{1} - \chi_{0}^{2} \|_{V}
           \quad \forall t \in
         [0,T].$$

Our second continuous dependence results holds in the more general frame of assumptions of Proposition \[prop:regularized\], but for more regular initial data. Indeed, we have

\[th:3.2\] Assume that holds for some $p\in
[0,1]$, and that $\phi$ complies with , , and . Let $\chi_{0}^{1}$ and $\chi_{0}^{2} $ be two initial data for Problem \[p:2\] such that $\chi_{0}^{i} \in Z$ and $\widehat{\phi}(\chi_{0}^{i}) \in L^1
    (\Omega)$ for $i=1,2$, and let $ \chi_{i} $, $ i=1,2 $, be the corresponding solutions. Then, for every $\delta>0$, there exists a positive constant $ S_{\delta} $, also depending on $T$, $|\Omega|$, $C_1$, $C_2$ and $M^{*}:= \max_{i=1,2} \{\| \chi_{0}^{i}\|_Z \}$, such that estimate holds for all $t \in [0,T]$.

Global attractor and exponential attractors for the viscous problem {#ss:2.4}
-------------------------------------------------------------------

The *energy functional* associated with Problem \[p:2\] reads $$\label{e:ene-funct} {\mathcal{E}}: {X}\to {\mathbb{R}}, \ \  \ \ {\mathcal{E}}(v):= \frac12
\int_{\Omega} |\nabla v|^2 + \int_{\Omega} \widehat{\phi}(v) \ \
\text{for all $v \in {X}$.}$$ Consequently, we introduce the phase space $({X},{\operatorname{d}_X})$ of energy bounded solutions, defined by $$\label{e:pspace}
\begin{aligned}
& {X}= \left\{v \in V\, : \ \widehat{\phi}(v) \in L^1 (\Omega)
\right\},
\\
&  {\operatorname{d}_X}(v_1,v_2) = \| v_1 -v_2 \|_{H^1 (\Omega)} +
 \left\|\widehat{\phi}(v_1) -
{\widehat{\phi}(v_2)} \right\|_{L^1(\Omega)} \qquad \text{for all
$v_1, \, v_2 \in {X}$.}
\end{aligned}$$

The following definition details the properties of the solutions to Problem \[p:2\] to which our long-time analysis will apply.

\[def:solp2\] We say that a function $\chi : [0,+\infty) \to
{X}$ is a *solution to Problem \[p:2\] on $(0,+\infty)$* if, for all $T>0$, $\chi$ enjoys regularity  on the interval $(0,T)$ and there exists a function $w$, with regularity  for all $T>0$, such that equations – hold almost everywhere on $\Omega \times (0,+\infty)$. We set $$\label{e:gen-semiflow} \mathcal{S}= \left \{ \chi : [0,+\infty ) \to
X \, : \ \text{$\chi$ is a solution to Problem~\ref{p:2} on
$(0,+\infty)$} \right\}\,.$$

We assume that, besides , $\alpha$ complies with the following condition, slightly stronger than : $$\label{e:hyp-add-alpha} \tag{H9}
 \exists\, \mathsf{c}_{\alpha}>0, \ \
\exists\, \Psi: {\mathbb{R}}\to [0,+\infty) \ \text{convex}\,:  \ \ \forall
\, r \in {\mathbb{R}}\, \quad \alpha(r) r - \mathsf{c}_{\alpha}|r|^{2p+2} =
\Psi(r)\,.$$ Hence, our first result asserts that the solution set $\mathcal{S}$ is a *generalized semiflow* in the sense of Definition \[def:generalized-semiflow\].

\[prop:2.1\] Assume , –. Then,

1.  every $\chi  \in \mathcal{S}$ (cf. ) complies with the *energy identity* $$\label{e:enid} \delta\int_s^t \int_{\Omega} |\chi_t|^2 + \int_s^t
    \int_{\Omega} \alpha'(w)|\nabla w|^2 + {\mathcal{E}}(\chi(t)) = {\mathcal{E}}(\chi(s))
    \quad \text{for all $0 \leq s \leq t$,}$$ the function $w:(0,+\infty) \to V$ being defined by  on $\Omega \times (0,+\infty)$.

2.  Assume that $\alpha$ in addition complies with . Then, the set $\mathcal{S}$ is a generalized semiflow in the phase space , and its elements are continuous functions from $[0,+\infty)$ onto $X$.

We prove our main result on the long-time behavior of the solutions to Problem \[p:2\] under a further condition on $\phi$, which in particular implies (and thus replaces) , namely $$\label{lim-infty-phi} \tag{H10}
\begin{aligned}
\lim_{r\to +\infty}\phi(r)=+\infty, \qquad \lim_{r\to
-\infty}\phi(r)=-\infty\,,
\\
\lim_{r\to +\infty}\phi'(r)=\lim_{r\to -\infty}\phi'(r)=+\infty\,.
\end{aligned}$$

\[th:4\] Assume , , , and . For a given $\mathrm{m}_0
>0$, denote by $\mathcal{D}_{\mathrm{m}_0}$ the set $$\label{e:fixed-mean-value} \mathcal{D}_{\mathrm{m}_0}= \left \{ \chi
\in X\, : \ |m(\chi)| \leq {\mathrm{m}_0} \right\}\,.$$ Then, the semiflow $\mathcal{S}$ possesses a unique global attractor $\mathcal{A}$ in $\mathcal{D}_{\mathrm{m}_0}$, given by $$\label{e:att} \mathcal{A}: =\bigcup\left \{\omega(D)\, : \ D \subset
\mathcal{D}_{\mathrm{m}_0} \ \text{bounded}\right\}\,.$$ Finally, we have the following enhanced regularity for the elements of the $\omega$-limit of every trajectory: $$\label{e:enhanced-regularity} \forall\, p \in [1,+\infty) \ \
\exists\,C_p>0\,: \ \ \forall\,\chi \in \mathcal{S}, \ \ \forall\,
\bar{\chi} \in \omega(\chi) \quad \| \bar{\chi}\|_{W^{2,p}(\Omega)}
+ \| \widehat{\phi}(\bar{\chi})\|_{L^p(\Omega)} \leq C_p\,.$$

\[rem:poly-advantages\] Notice that, in the case $$\label{e:special-case}
 \text{
 $\widehat{\phi}$ is a polynomial  of even degree $\mathsf{m} \geq 4$, with a positive leading
 coefficient,}$$ then conditions and are satisfied.

\[zeta\] In addition to hypotheses , , , and  of Theorem \[th:4\], assume that $\phi$ complies with . Then, the enhanced regularity estimate  holds for $\chi$.

This regularity is reflected in the further regularity $$\label{e:further-reg-att}
 \mathcal{A}\subset Z,$$ for the global attractor $\mathcal{A}$, which holds provided that one works with the (smaller) set of solutions to Problem \[p:2\] arising from the approximation procedure which will be detailed in Appendix. In fact, the estimates leading to  can be rigorously justified only for this approximate problem, as we will see in the proof of Proposition \[prop:regularized\], cf. Section \[s:3.3\]. Now, the aforementioned family of “approximable” solutions to Problem \[p:2\] (see, e.g., [@BabinVishik92; @rossi-segatti-stefanelli08; @schimperna07; @segatti06] for analogous constructions) complies with the properties defining a generalized semiflow, except for the concatenation axiom. This has motivated the introduction in [@rossi-segatti-stefanelli08; @segatti06] of the (slightly more general) notion of *weak* global attractor, tailored to the *weak* generalized semiflows without the concatenation property. Hence, relying on the abstract results of [@rossi-segatti-stefanelli08; @segatti06] and arguing as in the proof of Theorem \[th:4\], it is straightforward to prove that the semiflow associated with the approximable solutions to Problem \[p:2\] admits a *weak* global attractor for which  holds. On the other hand, Theorem \[th:5\] below shows that, under the stronger assumptions of Theorem \[th:3.2\], the semiflow possesses the standard global attractor $\mathcal{A}$ satisfying , namely, $\mathcal{A}$ is a compact and invariant set which attracts (in the $V$-metric) all bounded sets of initial data as time goes to infinity.

We conclude this section by showing that it is also possible to construct an exponential attractor through the short-trajectories approach developed in [@malek-prazak]. Let us first set for a given $\tau>0$ $$X_\tau=L^2(0,\tau;V),\quad Y_\tau=\left\{u\in L^2(0,\tau;Z)\,:\, u_t \in L^2(0,\tau;H)\right\}$$ and observe that $Y_\tau$ is compactly embedded in $X_\tau$.

Under assumptions , , , and , we know that, for any $\chi_0\in V$ and any $T>0$, there exists a pair $(\chi,w)$ which solves Problem \[p:2\] with the regularity , , , (cf. Theorem \[th:2\] and Proposition \[prop:regularized\]). In particular, $\chi\in Y_T$. In addition, thanks to  and arguing in the same way as in the forthcoming Section \[s:3.1\], it is not difficult to show that $\Vert
\chi\Vert_{Y_T}$ can be estimated uniformly with respect to $\Vert\chi_0\Vert_V$, . The energy identity also entails the existence of a bounded set $B^0\subset V$ such that, if $(\chi,w)$ is a solution to Problem \[p:2\] with the aforementioned properties, then there exists $t_0>0$, only depending on $\Vert\chi_0\Vert_V$, such that $\chi(t)\in B^0$ for all $t\geq
t_0$ and $\chi(t)\in B^0$ for all $t\geq 0$ whenever $\chi_0\in B^0$ (see the proof of the eventual boundedness of $\mathcal{S}$ in Section \[ss:5.2\]). Let us now consider the set $\mathcal{X}_\ell=\{\chi : (0,\ell)\to V\}$ of all the $\ell$-trajectories $\chi$ such that $(\chi,w)$ is a solution to Problem \[p:2\] satisfying , , , . Then, we endow this set with the $X_\ell$-topology (note that it might be a non-complete metric space). Moreover, denoting by $V_w$ the space $V$ endowed with the weak topology, we have $\mathcal{X}_\ell\subset
C^0([0,\ell];V_w)$. Consequently, any $\ell$-trajectory makes sense pointwise.

From now on, we assume that assumption holds for some $p\in[0,1]$. Thanks to , for any $\ell$-trajectory, there exists $\tau\in (0,\ell)$ such that $\chi(\tau)\in Z$. This is sufficient to conclude that $\chi$ is unique from $\tau$ on, as a consequence of Proposition \[prop:regularized\] and Theorem \[th:3.2\]. Therefore, if $\chi\in \mathcal{X}_\ell$ and $T>\ell$, then there exists a unique $\tilde \chi\in\mathcal{X}_T$ such that $\tilde\chi\vert_{[0,\ell]} = \chi$. Thus, we can define a semigroup $L_t$ on $\mathcal{X}_\ell$ by setting $$(L_t\chi)(\tau):= \tilde\chi(t+\tau),\qquad \tau\in [0,\ell],$$ where $\tilde\chi$ is the unique element of $\mathcal{X}_{\ell+\tau}$ such that $\tilde\chi\vert_{[0,\ell]} = \chi$.

Let us now set $$B^0_\ell:=\left\{\chi\in\mathcal{X}_\ell\,:\, \chi(0)\in B^0\right\}.$$ Then, by Proposition \[prop:regularized\], we can infer that the set $\left\{\chi\vert_{[\ell/2,\ell]}\,:\, \chi\in B^0_\ell\right\}$ is bounded in $L^\infty(\ell/2,\ell;Z)$. Hence, we can prove a continuous dependence estimate like , which allows us to apply [@malek-prazak Lemma 2.1] and deduce that $L_t$ is Lipschitz continuous on $B^0_\ell$, uniformly with respect to $t\in [0,\tau]$ for any fixed $\tau>0$. Observe that, arguing as in Section \[s:3.3\], we can prove that $B^1_\ell=\overline{L_\tau(B^0_\ell)}^{X_\ell} \subseteq B^0_\ell$ for some $\tau>0$. From this fact we deduce that the dynamical system $(\mathcal{X}_\ell,L_t)$ has a global attractor $\mathcal{A}_\ell$ (see [@malek-prazak Thm. 2.1]). In addition, $L_\tau: \mathcal{X}_\ell \to Y_\ell$ is Lipschitz continuous for some $\tau>0$. Indeed, recall that $B^1_\ell$ is bounded in $L^\infty(0,\ell;Z)\cap H^1(0,\ell;V)$ and use . Thus, on account of [@malek-prazak Thm. 2.2], we can infer that $\mathcal{A}_\ell$ has finite fractal dimension. In order to go back to the original geometric space $V$, we introduce the evaluation mapping $e: \mathcal{X}_\ell \to V,$ $e(\chi):=\chi(\ell)$. Then, we set $B^1:=e(B^1_\ell)$ and we note that, for any $\chi_0\in B^1$, there is a unique solution to Problem \[p:2\], so that the solution operator $S_t$ is well defined on $B^1$ and $S_t(B^1)\subseteq B^1$, for all $t\geq 0$. In addition, $e$ is (Lipschitz) continuous on $B^1_\ell$ (use and [@malek-prazak Lemma 2.1] once more). Therefore, we use [@malek-prazak Thm. 2.4] to deduce that $\mathcal{A}:=e(\mathcal{A}_\ell)$ is the finite-dimensional global attractor of the dynamical system $(B^1,S_t)$.

It remains to prove the existence of an exponential attractor. We already know that $L_t$ is Lipschitz continuous on $B^1_\ell$, uniformly with respect to $t\in [0,\tau]$ for every fixed $\tau>0$ (see above). Thus, we only need to show that $t\mapsto L_t\chi$ is Hölder continuous with values in $V$, uniformly with respect to $\chi\in B^1_\ell$. This follows from [@malek-prazak Lemma 2.2], recalling that $B^1_\ell$ is, in particular, bounded in $H^1(0,\ell;V)$. Hence, $(\mathcal{X}_\ell,L_t)$ has an exponential attractor $\mathcal{E}_\ell$ and $\mathcal{E}:=e(\mathcal{E}_\ell)$ is an exponential attractor for $(B^1,S_t)$.

Summing up, we have proved the

\[th:5\] Assume that holds for some $p\in[0,1]$. Also, assume , , and . Then, there exists a bounded invariant set $B^1\subset V$ such that Problem \[p:2\] generates a dynamical system $(B^1,S_t)$ which possesses an exponential attractor $\mathcal{E}$. In addition, the system also has a global attractor $\mathcal{A}$ with finite fractal dimension.

Note that, in the framework of Theorem \[th:5\], neither assumption  nor are needed.

Proofs of Theorems \[th:1\] and \[th:2\] {#s:3}
========================================

#### Scheme of the proofs of Theorems \[th:1\] and \[th:2\].

We will prove Theorems \[th:1\] and \[th:2\] by taking the limit of a suitable approximation scheme for Problems \[p:1\] and \[p:2\]. For the sake of readability, we postpone detailing such a scheme in Appendix.

In Section \[s:3.1\], we will instead perform all estimates leading to the aforementioned passage to the limit directly on systems – and –. Note that, at this stage, some of the following calculations will only be formal, cf. Remark \[solo-formale\] below. Their rigorous justification will be given in Appendix, see Section \[ss:a.1\].

Next, in Section \[s:3.2\] (in Section \[s:3.3\], respectively), we will carry out a passage to the limit in some unspecified approximation scheme for Problem \[p:1\] (for Problem \[p:2\], respectively) and conclude the (formal) proof of Theorem \[th:1\] (of Theorem \[th:2\], respectively). In Section \[ss:a.2\], we will adapt the limiting arguments developed in Sections \[s:3.2\] and \[s:3.3\] to the approximation scheme for Problems \[p:1\] and \[p:2\] and carry out the rigorous proofs of the related existence theorems.

\[not:3.1\] We will perform the a priori estimates on systems – and –, distinguishing the ones which hold both in the viscous and the non-viscous cases from the ones which depend on the constant $\delta$ in  (which can be either strictly positive or equal to zero), and on our different assumptions on the nonlinearity $\phi$ in the viscous and non-viscous cases. Accordingly, we will use the generic notation $C$ for most of the constants appearing in the forthcoming calculations and depending on the problem data, and $C_\delta$ ($C_0$, respectively) for those constants *substantially* depending on the problem data and on $\delta >0$ (on $\delta =0$, respectively). We will adopt the same convention for the constants $S^i$, $S^i_\delta$, $S_0^i$, $i\geq 1$.

A priori estimates {#s:3.1}
------------------

#### First a priori estimate.

We test  by $w$, by $\chi_t$, add the resulting relations, and integrate over some time interval $(0,t) \subset (0,T)$. Elementary calculations lead to $$\label{est:1} \int_0^t \int_{\Omega} \alpha'(w) |\nabla w|^2 +
\delta \int_0^t \| \chi_t \|_{H}^2 + \frac12 \| \nabla \chi(t)
\|_{H}^2 + \int_{\Omega} \widehat{\phi} \left( \chi(t)\right)
=\frac12 \| \nabla \chi_0 \|_{H}^2 + \int_{\Omega} \widehat{\phi}
\left( \chi_0\right)\,.$$ Recalling , the second of  (which, in particular, yields that $\alpha'$ is bounded from below on ${\mathbb{R}}$ by a positive constant) and the positivity of $\widehat{\phi}$ (cf. ), we conclude that, for some constant $S^1
>0$, $$\label{e:stima1} \| \nabla w \|_{L^2 (0,T;H)} + \| \nabla \chi
\|_{L^\infty (0,T;H)} + \| \widehat{\phi}(\chi)\|_{L^\infty (0,T;L^1
(\Omega))} \leq S^1.$$

#### First a priori estimate in the viscous case.

In the case $\delta>0$, from the previous a priori estimate we also have $$\label{e:stima1-delta}  \| \chi_t \|_{L^2 (0,T;H)} + \|
\chi\|_{L^\infty (0,T;V)} \leq S_\delta^1\,.$$

#### Second a priori estimate.

We test  by $1$ and find ${m}(\chi_t)=0$ a.e. in $(0,T)$, so that, in particular, $$\label{e:constant-m-value}
 {m}(\chi(t))=m_0:={m}(\chi_0) \qquad \forall\, t \in [0,T].$$ Hence, testing  by $1$, we obtain $$\label{est:2} {m}(\phi(\chi(t))) = {m}(w(t)) \qquad {\text{for a.a.}}\  t \in
(0,T).$$

#### Second a priori estimate in the non-viscous case.

It follows from and the Poincaré inequality that $$\label{est:2-deltazero} \| \chi\|_{L^\infty (0,T;V)} \leq S^2\,.$$

#### Third a priori estimate in the non-viscous case.

We test  by $\chi-{m}(\chi)$: we have, for a.e. $t \in
(0,T)$, $$\label{est:3}
\begin{aligned}
 \| \nabla \chi(t)  \|_{H}^2 +
\int_{\Omega} \phi(\chi(t)) \left( \chi(t) - {m}(\chi(t)) \right) &=
\int_{\Omega} w(t) \left( \chi(t) - {m}(\chi(t)) \right)  \\ & =
\int_{\Omega} \left( w(t) - {m}(w(t)) \right) \left( \chi(t) -
{m}(\chi(t)) \right)\\ &  \leq  C \| \nabla \chi(t) \|_{H} \| \nabla
w(t) \|_{H} \leq C S^1 \| \nabla w(t) \|_{H} ,
\end{aligned}$$ the latter estimate ensuing from the Poincaré inequality for zero mean value functions and the previous . On the other hand, and  yield that there exist constants $C_{m_0}, C_{m_0}'>0$ such that, for a.e. $t \in [0,T]$, $$\label{e:useful-later} \int_{\Omega} |\phi( \chi(t))| \leq C_{m_0}
\int_{\Omega} \phi(\chi(t)) \left( \chi(t) - {m}(\chi(t)) \right) +
C_{m_0}' \,.$$ Combining this with , we deduce that there exists $C>0$, also depending on $C_{m_0}$ and on $C_{m_0}'$, such that, for a.e. $t \in (0,T)$, $$\label{est:4}
 \int_{\Omega} |\phi( \chi(t))| \leq C \left(\| \nabla
w(t) \|_{H} + 1\right)\,.$$ Thus, in view of , we obtain an estimate for $\phi(\chi)$ in $L^2
(0,T; L^1 (\Omega))$. Finally, due to , we find $$\| {m}(w) \|_{L^2 (0,T)} \leq C_0.$$ Hence, by  and the Poincaré inequality, we conclude that $$\label{e:stima3} \| w \|_{L^2 (0,T;V)} \leq S^1_0\,.$$

#### Third a priori estimate in the viscous case.

Estimate  for $\widehat{\phi}(\chi)$ and  yield that $$\label{est:3-delta} \| \phi(\chi) \|_{L^\infty (0,T; L^1 (\Omega))
 } \leq S_\delta^2\,.$$ Recalling , we immediately infer that $$\label{est:4-delta} \| {m}(w) \|_{L^\infty (0,T)} \leq S_\delta^3,$$ whence, again, $$\label{est:5-delta} \| w \|_{L^2 (0,T; V)} \leq S_\delta^4.$$

#### Fourth a priori estimate in the non-viscous case.

We preliminarily observe that, thanks to , equation  can be rewritten as $$\label{e:more-convenient_form} A\chi + \beta(\chi) = w + C_{\phi,1}
\chi \qquad {\text{a.e.\ in}}\ \Omega \times (0,T)\,.$$ Notice that, in view of  and , the right-hand side of  belongs to $L^2 (0,T; L^6 (\Omega))$. Hence, we can test  by $|\beta(\chi)|^4 \beta(\chi)$ and easily conclude that $$\|A\chi\|_{L^2 (0,T; L^6 (\Omega))} +\|\beta(\chi)\|_{L^2 (0,T;
L^6 (\Omega))} \leq C_0\,.$$ Then, also by standard elliptic regularity results, we find $$\label{e:stima3-bis} \| \phi(\chi)\|_{L^2 (0,T; L^6 (\Omega))} +  \|
\chi \|_{L^2 (0,T;W^{2,6} (\Omega))} \leq S^2_0\,.$$

#### Fourth a priori estimate in the viscous case.

We combine  and and argue by comparison in . Relying on  and on the related elliptic regularity estimate, we have $$\label{e:useful-again} \| \phi(\chi)  \|_{L^2 (0,T;H)} \leq
S^5_\delta\,,$$ as well as an estimate for $A\chi$ in $L^2 (0,T; H)$, so that $$\label{e:stima3-delta} \| \chi \|_{L^2 (0,T;Z)} \leq S^6_\delta\,.$$

#### Fifth a priori estimate.

It follows from  and  that $\int_0^T \int_{\Omega} w^{2p} |\nabla w|^2
\leq C$, whence we conclude that $$\label{est:3.14} \| \nabla (|w|^{p}w) \|_{L^2 (0,T;H)} \leq S^3\,.$$

#### Sixth a priori estimate in the non-viscous case.

From , , and , we deduce that $$\label{e:crucial0}
 \||\phi(\chi)|^{\sigma}\|_{L^{2/\sigma}(0,T; L^{6/\sigma} (\Omega)) \cap L^\infty (0,T; L^1
 (\Omega))} \leq C_0\,.$$ Using the interpolation inequality $$\forall\, v \in L^1 (\Omega) \cap L^{6/\sigma} (\Omega) \quad  \| v
\|_{L^{1/\sigma} (\Omega)} \leq \| v \|_{L^1 (\Omega)}^{\theta}\, \| v
\|_{L^{6/\sigma} (\Omega)}^{1-{\theta}}, \quad \text{with ${\theta}=
\frac{5\sigma}{6-\sigma}$,}$$ we obtain the estimate $$\label{e:crucial}
 \||\phi(\chi)|^{\sigma}\|_{L^{q_\sigma}(0,T; L^{1/\sigma}
 (\Omega))} \leq C_0, \quad
  \text{with $q_\sigma=\frac{2}{\sigma} \, \frac{1}{1-{\theta}}= \frac{6-\sigma}{3\sigma-3\sigma^2}$,}$$ whence a bound for $\phi(\chi)$ in $L^{\sigma q_\sigma} (0,T; L^1
(\Omega))$. Taking into account , we conclude that $$\label{e:3.15} \| {m}(w) \|_{L^{\sigma q_\sigma}(0,T)} \leq C_0\,,
\quad \text{whence} \quad \| |{m}(w)|^{p+1} \|_{L^{(\sigma
q_\sigma)/(p+1)}(0,T)} \leq C_0\,.$$ On the other hand, applying the *nonlinear* Poincaré inequality  with the choices $X=V$, $Y=H$, $Gv= \nabla v$, and $\Psi(v)= |\Omega|^{-p-1}|\int_{\Omega} |v|^{\frac{1}{p+1}} \mathrm{sign}(v)|^{p+1}$, where $v=|w|^p w$, we find $$\label{e:poinc-est} \||w|^p w  \|_{V} \leq K \left( \| \nabla
(|w|^{p}w) \|_{H} + \left| {m}(w)\right|^{p+1}\right)\,.$$ Therefore, combining estimate  for $|{m}(w)|^{p+1}$ with , we finally obtain, owing to the Poincaré inequality , $$\label{e:to-be-cited3} \| |w|^p w\|_{L^{(\sigma
q_\sigma)/(p+1)}(0,T; V)} \leq C_0.$$ Using the embedding $V \subset L^6
(\Omega)$ and the growth  for $\alpha$, we infer

\[e:est-refined\] $$\label{e:est-refined_1}
 \| \alpha(w) \|_{L^{\eta_{p\sigma}} (0,T; L^{\kappa_p} (\Omega))}
 \leq S_0^3\,$$ (where we have used the fact that $(\sigma q_\sigma)/(2p+1)$ equals the index $\eta_{p\sigma}$ defined in ). Hence, by comparison in , we also conclude that $$\label{e:3.18} \| \chi_t \|_{L^{\eta_{p\sigma}}} (0,T;
{\mathcal{W}^{-2,{\kappa_p}}(\Omega)}) \leq S_0^4$$

(see again  for the definition of $\kappa_p$).

#### Sixth a priori estimate in the viscous case.

Combining , and the Poincaré-type inequality , we deduce an estimate for $|w|^{p}w $ in $L^2 (0,T;V)$. Then, arguing in the same way as for , we have $$\label{e:to-be-quoted}
 \| \alpha(w) \|_{L^{\rho_p} (0,T; L^{\kappa_p}
(\Omega))} \leq C_\delta,$$ the index $\rho_p$ being defined in . Now, in view of estimate  for $\chi_t$ in $L^2 (0,T;H)$, a comparison in  yields an estimate for $A(\alpha (w))$ in $L^2 (0,T;H)$. By elliptic regularity results, we finally conclude that $$\label{e:3.19}
 \| \alpha(w) \|_{L^{\rho_p} (0,T; Z)} \leq S_\delta^7\,.$$

#### Seventh a priori estimate in the non-viscous case.

Our aim is now to show that $$\label{useful2}
 \| \phi(\chi) \|_{L^{\sigma q_\sigma} (0,T;L^{6}(\Omega))} \leq S_0^5, \qquad \text{with $\sigma q_\sigma =
\frac{6-\sigma}{3-3\sigma}>2$\,.}$$ Indeed, again recalling the embedding $V \subset L^6 (\Omega)$, we observe that yields an estimate for $w$ in $L^{\sigma q_\sigma}(0,T; L^{6p+6}(\Omega))$. Then, taking into account estimate  for $\chi$ in $L^\infty
(0,T; L^6 (\Omega))$, together with the aforementioned elliptic regularity argument, we find estimate  by a comparison in .

#### Seventh a priori estimate in the viscous case.

We combine estimate , the continuous embedding $Z \subset
 L^\infty(\Omega)$, and the growth condition  to deduce an estimate for $w$ in $L^{\rho_p (2p+1)} (0,T;L^\infty (\Omega))$, whence $$\label{e:3.22} \| w\|_{L^{2p+2} (0,T; L^{\infty}(\Omega))} \leq
S_\delta^8\,.$$

\[solo-formale\] Notice that all the a priori estimates for the viscous Problem \[p:2\] are in fact rigorously justified on system –. This has significant repercussions on the long-time analysis of Problem \[p:2\]. Indeed, this allows us to work with the semiflow associated with the solutions to Problem \[p:2\] (cf. ) and prove the existence of a global attractor in the sense of [@Ball97]. However, as pointed out in Remark \[zeta\], if we address further regularity properties of the attractor (e.g., ), then we need additional estimates which cannot be performed directly on system –, due to insufficient regularity of the solutions. Thus, we have to rely on some approximation. On the one hand, this leads to a smoother attractor $\mathcal{A}$, but, on the other hand, we lose the concatenation property of the trajectories (cf. [@rossi-segatti-stefanelli08; @segatti06]); moreover, only trajectories which are limits of the approximation scheme will be attracted by the smoother attractor $\mathcal{A}$.

We also point out that the viscous system – cannot be used as an approximation for the non-viscous problem. Indeed, it is not difficult to realize that the fourth a priori estimates – (yielding a bound for $\phi(\chi)$ which plays a crucial role in the ensuing calculations) are not compatible with the term $\delta\chi_t$ in . This fact seems to suggest the use of two different approximation schemes for Problem \[p:1\] and Problem \[p:2\], which would lead to cumbersome and repetitious calculations. In order to circumvent this problem, we will construct in Appendix an approximation scheme depending on two distinct parameters and prove the existence of solutions to Problem \[p:1\] by passing to the limit in three steps. Since the (rigorous) proof of existence for Problem \[p:2\] can be performed along the very same lines, we have chosen not to detail it in Appendix.

Proof of Theorem \[th:1\] {#s:3.2}
-------------------------

Let $\{ (\chi_n, w_n )\} $ be some sequence of approximate solutions to Problem \[p:1\]. Due to estimates , , , , and , applying standard compactness and weak compactness results (see [@simon]), we find that there exists a pair $(\chi,w)$ with the regularities specified by – such that, along a (not relabeled) subsequence, the following strong, weak, and weak$^*$ convergences hold as $n \to +\infty$: $$\begin{aligned}
 \label{e:conv-chi-strong}
 &
\begin{aligned}
  \chi_n \to \chi \quad &\text{in \ $L^2
(0,T; W^{2-{\varepsilon},6} (\Omega)) \cap L^q (0,T;V) \cap {\mathrm{C}}^0 ([0,T];
H^{1-{\varepsilon}} (\Omega))$} \\ &  \text{for  every ${\varepsilon}>0 $ and $1 \leq
q < +\infty$,}
\end{aligned}
\\
& \label{e:conv-chi-weak1}
 \chi_n {{\stackrel{*}{\rightharpoonup}\,}}\chi \quad  \text{in \ $L^2
(0,T; W^{2,6} (\Omega)) \cap L^\infty (0,T;V)$,} \\
& \label{e:conv-chi-weak2}
  \chi_{n,t}
{\rightharpoonup}\chi_t \quad \text{in \ $L^{\eta_{p\sigma}} (0,T;
{\mathcal{W}^{-2,\kappa_p}(\Omega)})$,}
\\
& \label{e:conv-w-strong} w_n  {\rightharpoonup}w \quad \text{in \ $L^2 (0,T;
V)$.}\end{aligned}$$ Furthermore, there exists $\bar{\alpha} \in L^{\eta_{p\sigma}} (0,T;
L^{\kappa_p} (\Omega))  $ such that $$\label{e:conv-alpha-weak} \alpha(w_n) {\rightharpoonup}\bar{\alpha} \quad
\text{in \ $L^{\eta_{p\sigma}} (0,T; L^{\kappa_p} (\Omega))$.}$$ Now, estimate in particular yields (recall that $\sigma q_\sigma >2$) that $$\label{e:cruc-pass1} \text{the sequence $\{ \phi(\chi_n)\}$ is
uniformly integrable in $L^2 (0,T;H)$.}$$ Furthermore, we have, up to a further subsequence, $$\label{e:cruc-pass2} \phi(\chi_n(x,t)) \to \phi(\chi(x,t)) \qquad
{\text{for a.a.}}\, (x,t) \in \Omega \times (0,T)\,,$$ which is a consequence of the continuity of $\phi$ and of the pointwise convergence (up to a further subsequence) $\chi_n(x,t) \to \chi(x,t)$ a.e. in $\Omega \times (0,T)$ (cf. ). Combining  and  and recalling the compactness criterion Theorem \[t:ds\], we conclude that $$\label{e:strong-phi} \phi(\chi_n) \to \phi(\chi) \qquad \text{in \
$L^2 (0,T; H). $}$$ Exploiting –, one easily concludes that the triplet $(\chi,w,\bar{\alpha})$ satisfies $$\begin{aligned}
& \label{1-var-lim} \chi_t + A \bar{\alpha} =0 \qquad \text{in
${\mathcal{W}^{-2,\kappa_p}(\Omega)}$} \quad {\text{a.e.\ in}}\ (0,T)\,,
\\
& \label{2-var-lim} A\chi + \phi(\chi) =w \qquad {\text{a.e.\ in}}\ \Omega
\times (0,T)\,.\end{aligned}$$ Finally, in order to prove that $$\label{final-step} \bar{\alpha} (x,t) = \alpha (w(x,t)) \qquad
{\text{for a.a.}}\ (x,t) \in \Omega \times (0,T)\,,$$ we test the equation approximating by $w_n$ and integrate in time. We thus have $$\begin{aligned}
\lim_{n \to +\infty}\int_0^T \int_{\Omega} |w_n|^2 &= \lim_{n \to
+\infty} \int_0^T \int_{\Omega}\phi(\chi_n) w_n + \lim_{n \to +\infty}
\int_0^T \int_{\Omega}\nabla \chi_n \cdot \nabla w_n \\ & = \int_0^T
\int_{\Omega}\phi(\chi) w+ \int_0^T \int_{\Omega}\nabla \chi
\cdot\nabla w\\ & =\int_0^T \int_{\Omega} |w|^2,
\end{aligned}$$ where the second equality follows from convergences , , and , and the last one from . Hence, we conclude that $$w_n \to w \ \  \text{in $L^2 (0,T;H)$, \ whence} \quad w_n \to w \ \
{\text{a.e.\ in}}\ \Omega \times (0,T)$$ (the latter convergence holding up to a subsequence). By continuity of $\alpha$, we also have $\alpha(w_n) \to \alpha (w)$ a.e. in $\Omega \times (0,T)$. Estimate  (recall ) and again Theorem \[t:ds\] yield, for instance, that $$\alpha(w_n) \to \alpha(w) \quad \text{in $L^1 (0,T; L^1 (\Omega))$,}$$ whence the desired equality .

Proofs of Theorem \[th:2\] and Proposition \[prop:regularized\] {#s:3.3}
---------------------------------------------------------------

#### Proof of Theorem \[th:2\].

Let $\{ (\chi_n, w_n )\} $ be some sequence of approximate solutions to Problem \[p:2\]. Thanks to estimates , , , , , and , applying standard compactness and weak compactness results (see [@simon]), we find a triplet $(\chi,w,\bar{\alpha})$ such that, along a (not relabeled) subsequence, the following strong, weak, and weak$^*$ convergences hold as $n \to +\infty$: $$\begin{aligned}
 \label{e:conv-chi-strong-v}
 &
\begin{aligned}
  \chi_n \to \chi \quad &\text{in \ $L^2
(0,T; H^{2-{\varepsilon}} (\Omega)) \cap L^q (0,T;V) \cap {\mathrm{C}}^0 ([0,T];
H^{1-{\varepsilon}} (\Omega))$} \\ &  \text{for  every ${\varepsilon}>0 $ and $1 \leq
q < +\infty$,}
\end{aligned}
\\
& \label{e:conv-chi-weak-v}
\begin{aligned}
  \chi_n {{\stackrel{*}{\rightharpoonup}\,}}\chi \quad &  \text{in \ $L^2
(0,T; Z) \cap L^\infty (0,T;V) \cap H^1 (0,T;H)$,}
\end{aligned}
\\
& \label{e:conv-w-strong-v} w_n  {{\stackrel{*}{\rightharpoonup}\,}}w \quad \text{in \ $L^2
(0,T; V) \cap L^{2p+2} (0,T; L^{\infty}(\Omega))$.}\end{aligned}$$ In particular, from , we deduce that $$\label{step-1} {\mathcal{N}}(\chi_{n,t}) {\rightharpoonup}{\mathcal{N}}(\chi_t) \qquad \text{in
$L^2 (0,T; Z)$.}$$ Furthermore, by , there exists $\bar{\alpha}
\in L^{\rho_p} (0,T; L^{\kappa_p} (\Omega)) $ such that $$\label{e:conv-alpha-weak-v} \alpha(w_n) {\rightharpoonup}\bar{\alpha} \quad
\text{in \ $L^{\rho_p} (0,T; L^{\kappa_p} (\Omega))$.}$$ Now, up to a subsequence, by the last of  and by continuity of $\phi$, we have, for all $t \in [0,T]$, $$\label{step0} \phi(\chi_n (\cdot,t)) \to \phi(\chi(\cdot,t)) \qquad
{\text{a.e.\ in}}\  \Omega\,.$$ On the other hand, it follows from estimate  that $$\label{step00}
  \text{the sequence $\{ \phi(\chi_n )\}$ is
uniformly integrable in $L^1 (0,T;L^1 (\Omega))$.}$$ Then, by – and Theorem \[t:ds\], we conclude that, along the same subsequence as in , $\phi(\chi_n) \to
 \phi(\chi)$ in $L^1 (0,T; L^1 (\Omega))$. We then have, up to a subsequence, $$\label{step1} \phi(\chi_n (t)) \to \phi(\chi(t)) \qquad \text{in
$L^1 (\Omega)$}  \ \ {\text{for a.a.}}\ t \in (0,T)\,.$$ Next, using , we see that $\phi(\chi_n)$ is uniformly integrable in $L^{\nu} (0,T; L^1 (\Omega))$ for all $\nu \in [1,+\infty)$. Applying Theorem \[t:ds\], from , we deduce that $$\label{step2} \phi(\chi_n ) \to \phi(\chi) \qquad \text{in $L^{\nu}
(0,T; L^1(\Omega))$ \ for every $\nu \in [1,+\infty)$.}$$ Collecting – and , we conclude that the triplet $(\chi,w,\bar{\alpha})$ satisfies $$\begin{aligned}
& \label{1-var-lim-v} \chi_t + A \bar{\alpha} =0 \qquad {\text{a.e.\ in}}\
\Omega \times (0,T)\,,
\\
& \label{2-var-lim-v} \delta \chi_t + A\chi + \phi(\chi) =w \qquad
{\text{a.e.\ in}}\  \Omega \times (0,T)\,.\end{aligned}$$ It remains to show that $\bar{\alpha} \equiv \alpha(w)$. To this aim, we note that $\alpha$ defines a maximal monotone graph in the duality $(L^{2p+2} (\Omega\times (0,T)), L^{\rho_p} (\Omega\times
(0,T)))$ (note that $\rho_p$ and $2p+2$ are conjugate exponents). Taking into account relations and , and applying a well-known result from the theory of maximal monotone operators in Banach spaces (see [@barbu Lemma 1.3, p. 42]), it is then sufficient to prove that $$\label{3.40} \limsup_{n\to +\infty} \int_{0}^T \int_{\Omega}
\alpha(w_n) w_n \leq \int_{0}^T \int_{\Omega} \bar{\alpha} w\,.$$ Now, $$\label{a1}
\begin{aligned}
\int_{0}^T \int_{\Omega} \alpha(w_n) w_n & =\int_{0}^T \int_{\Omega}
\big(\alpha(w_n)-{m}(\alpha(w_n))\big)\, w_n +|\Omega| \int_{0}^T
{m}(\alpha(w_n))\, {m}(w_n)\\ & = -\int_{0}^T\int_{\Omega}
{w_n}\,{{\mathcal{N}}(\chi_{n,t})} +|\Omega| \int_{0}^T {m}(\alpha(w_n))\,
{m}(w_n)\,,
\end{aligned}$$ where the second equality follows from . Then, using  and , we find the chain of inequalities $$\label{e:3.40}
\begin{aligned}
 \liminf_{n \to +\infty}
& \Big( \int_{0}^T \int_{\Omega}{w_n}\,{{\mathcal{N}}(\chi_{n,t})}\Big)
\\
&  \geq  \liminf_{n \to +\infty} \delta \int_{0}^T \| \chi_{n,t}
\|_{V'}^2 + \lim_{n \to +\infty} \int_{0}^T \int_{\Omega} \chi_{n,t}
\chi_n  + \lim_{n \to +\infty} \int_{0}^T \int_{\Omega}\phi(\chi_n)
{\mathcal{N}}(\chi_{n,t})\\ & \geq \delta \int_{0}^T \| \chi_{t} \|_{V'}^2 +
\int_0^T \int_{\Omega} \chi_t \chi + \int_{0}^T
\int_{\Omega}\phi(\chi) {\mathcal{N}}(\chi_{t})\\ &= \int_{0}^T \int_{\Omega}
{w}\,{{\mathcal{N}}(\chi_{t})} =-\int_{0}^T \int_{\Omega}
\big(\bar{\alpha}-{m}(\bar{\alpha})\big)\, w\,,
\end{aligned}$$ where the second inequality follows from convergences  and  for $\chi_n$ and from combining  with , while the subsequent identities are due to –. On the other hand, it follows from  that $$\label{step3} {m}(\alpha(w_n)) {\rightharpoonup}{m}(\bar{\alpha}) \qquad
\text{in $L^{\rho_p} (0,T)$,}$$ whereas, from , we gather that $$\label{step4} {m}(w_n)= {m}(\phi (\chi_n)) \to
{m}(\phi(\chi))={m}(w) \qquad \text{in $L^{2p+2} (0,T)$.}$$ Combining –, we conclude that $$\label{e:3.41} \lim_{n \to +\infty} |\Omega| \int_{0}^T
{m}(\alpha(w_n))\, {m}(w_n)=
 |\Omega| \int_{0}^T {m}(\bar{\alpha})\, {m}(w)\,.$$ Collecting , , and , we infer the desired . Ultimately, we have proved that $$\label{e:for-later-convenience}
 \alpha(w_n) {\rightharpoonup}\alpha(w) \ \
\text{in \ $L^{\rho_p} (0,T; L^{\kappa_p} (\Omega))$  \ \ and} \ \
\lim_{n\to +\infty} \int_{0}^T \int_{\Omega} \alpha(w_n) w_n =
\int_{0}^T \int_{\Omega} \alpha(w) w\,.$$

#### Proof of Proposition \[prop:regularized\].

In order to prove that system – enjoys the regularization in time –, using the Gagliardo-Nirenberg interpolation inequality we note that $L^2(0,T; Z) \cap L^\infty (0,T;V) \subset L^8 (0,T; W^{1,12/5}(\Omega)) $ with continuous embedding. Therefore, regularity  for $\chi$ and standard Sobolev embeddings yield $$\label{e:stichi} \| \chi\|_{L^8 (0,T; L^{12}(\Omega))} \leq C\,.$$

Now, we test by $A\chi_t$. Note that all the forthcoming computations are rigorous on the approximation scheme for Problem \[p:2\] which we will detail in Appendix. Elementary calculations yield $$\label{e:elem1}
\begin{aligned}
    \frac{{\mathrm{d}}}{ {\mathrm{d}}t}\left(\frac12 \int_{\Omega}|A\chi|^2 \right) +
\delta\int_{\Omega}|\nabla \chi_t|^2 = I_1 + I_2\,,
\end{aligned}$$ with $$\begin{aligned}
&
 \label{e:elem2}    I_1  :=  \int_{\Omega} \nabla w
\,\cdot\, \nabla \chi_t \leq \frac{\delta}2\int_{\Omega} |\nabla
\chi_t|^2 + \frac{1}{2\delta}\int_{\Omega} |\nabla w|^2\,,
\\
&  \label{e:elem3}
\begin{aligned}
 I_2   &  := -  \int_{\Omega}  \phi{'}(\chi) \left(\nabla\chi
 \cdot \nabla \chi_t\right)
\\ & \leq C_{\phi,3} \int_{\Omega}|\nabla \chi||\nabla
\chi_t| \left( 1+|\chi|^4\right)\\ &  \leq C \| \nabla \chi_t \|_{H}
\| \nabla \chi \|_{L^6(\Omega)} \left(\| \chi\|_{L^{12}(\Omega)}^4 +
1 \right)\\ &  \leq \frac{\delta}{4}\| \nabla \chi_t \|_{H}^2 + C
\left(\| \chi\|_{L^{12}(\Omega)}^8 + 1 \right) \| \chi \|_{Z}^2\,,
\end{aligned}\end{aligned}$$ where the second inequality follows from , the third one from the Hölder inequality, and the last one by taking into account the continuous embedding $Z \subset W^{1,6}(\Omega)$.

Collecting –, taking into account , and applying the uniform Gronwall Lemma (see [@temam Lemma III.1.1]), we find for every $\tau>0$ an estimate of the form for $\nabla \chi_t $ in $L^2 (\tau,T;H) $ and for $A\chi $ in $L^\infty (\tau,T;H)$, whence $$\chi \in L^\infty (\tau,T;Z) \cap H^1 (\tau,T;V)\quad \text{for all
} 0<\tau<T\,.$$ Then, a comparison in  also yields a bound for $A(\alpha(w))$ in $L^{\rho_p}(\tau,T;V)$, whence an estimate for $\alpha(w) $ in $L^{\rho_p}(\tau,T;H^3(\Omega))$, in view of . Thus, we conclude –, as well as estimate .

Global attractor for Problem \[p:2\] {#s:5}
====================================

Proof of Proposition \[prop:2.1\] {#ss:5.1}
---------------------------------

We need two preliminary lemmas. The first one clarifies some properties of the energy functional ${\mathcal{E}}$ .

\[l:5.1\] Assume –. Then, the functional ${\mathcal{E}}: {X}\to {\mathbb{R}}$ defined by  is bounded from below, lower-semicontinuous w.r.t. the $H$-topology, and satisfies the chain rule $$\label{e:l5.1}
\begin{gathered}
\text{for all $v \in H^1(0,T;H)$ with $Av+ \phi(v)  \in
L^2(0,T;H)$,} \\
\text{ the map $t \in [0,T] \mapsto {\mathcal{E}}(v(t))$ is absolutely
continuous, and}
\\
\frac{\mathrm{d}}{\mathrm{d}t} {\mathcal{E}}(v(t)) = \int_{\Omega}  v_t(t)
\left(Av(t)+ \phi(v(t)) \right) \qquad {\text{for a.a.}}\ t \in (0,T)\,.
\end{gathered}$$

In order to prove the lower-semicontinuity property, we fix a sequence $\{v_n \}$ converging to some $v $ in $H$ and assume, without loss of generality, that $\sup_n {\mathcal{E}}(v_n) <+\infty$. Since $\widehat{\phi}$ is bounded from below, we conclude that $\{v_n\}$ is actually bounded in $V$, and thus $v_n {\rightharpoonup}v$ in $V$, yielding $\textstyle \int_{\Omega} |\nabla v|^2  \leq \liminf_{n}
\int_{\Omega} |\nabla v_n|^2$. On the other hand, $$\begin{aligned}
\liminf_{n \to +\infty} \int_{\Omega} \widehat{\phi}(v_n) &  =
\liminf_{n \to +\infty} \int_{\Omega}
\left(\widehat{\phi}(v_n)+\frac{C_{\phi,2}}{2} |v_n|^2 \right) -
\frac{C_{\phi,2}}2\lim_{n \to +\infty} \int_{\Omega}|v_n|^2  \\ &
\geq \int_{\Omega}\left( \widehat{\phi}(v)+\frac{C_{\phi,2}}2
|v|^2\right) -\frac{C_{\phi,2}}2\int_{\Omega}|v|^2\,, \end{aligned}$$ the latter inequality following from  and, for instance, from Ioffe’s Theorem \[th-ioffe\]. Finally, to check the chain rule , we observe that the functional $$\label{funz-bar} \mathcal{E}_{\mathrm{cv}} (v): = \mathcal{E}(v)  +
\frac{C_{\phi,2}}{2}\int_{\Omega} |v|^2 \qquad \text{for all $v \in
X$}$$ is convex, thanks to . Then, follows from the chain rule for $ \mathcal{E}_{\mathrm{cv}}$, see [@brezis73 Lemma III.3.3].

\[l:5.2\] Assume . Then, $$\label{e:5.2.2} \text{for all $w \in V \cap L^\infty (\Omega)$, there
holds} \ \ \nabla \alpha(w(x)) = \alpha'(w(x)) \nabla w(x) \ \
{\text{for a.a.}}\ x \in \Omega\,.$$

Since $\Omega$ is smooth, we can take a sequence $\{w_k \}\subset
\mathrm{C}^1 (\overline{\Omega})$ such that $w_k \to w$ in $V \cap
L^q (\Omega)$ for all $1\leq q<+\infty$. Clearly, for all $k \in {\mathbb{N}}$, there holds $$\label{e:clearl} \nabla \alpha(w_k(x)) = \alpha'(w_k(x)) \nabla
w_k(x) \qquad \forall\, x \in \Omega\,.$$ Now, since $\alpha'(r)$ grows like $|r|^{2p}$ by , we conclude that $\alpha'(w_k)\to \alpha'(w) $ and $\alpha(w_k)\to
\alpha(w) $ in $L^q (\Omega)$ for all $1\leq q<+\infty$. Therefore, $\nabla \alpha(w_k) = \alpha'(w_k) \nabla w_k \to \alpha'(w) \nabla
w $ in $L^{\rho}(\Omega)$ for all $\rho\in[1,2)$ and  follows.

#### Proof of Proposition \[prop:2.1\].

Thanks to Theorem \[th:2\], the set ${\mathcal{S}}$ complies with the existence axiom **(P1)** in Definition \[def:generalized-semiflow\]. The translation property **(P2)** is immediate to check. Concerning the concatenation axiom, let $\chi_1$ and $\chi_2$ be two solutions to Problem \[p:2\] on $(0,+\infty)$, satisfying $\chi_1(\tau) =
\chi_2 (0)$ for some $\tau \geq 0$, and let the functions $w_1$ and $w_2$ be such that, for $i=1,2$, the pairs $(\chi_i,w_i)$ satisfy equations –, with regularities  and . Then, one easily sees that the concatenations (cf. ) $\tilde
\chi$ and $\tilde w$ of $\chi_1, \, \chi_2$ and $w_1,\, w_2$, respectively, satisfy equations –, and still enjoy regularities  and , respectively (the fact that $\chi_1(\tau) = \chi_2 (0)$ is crucial for the time-regularity of $\tilde{\chi}$).

To prove that all solutions $\chi \in {\mathcal{S}}$ are continuous w.r.t. the phase space topology , let us fix $\{t_n\}, t_0$ in $[0,+\infty)$, and show that $$\label{e:continuity}
 t_n \to t_0 \ \Rightarrow \ \left( \| \chi(t_n)
-\chi(t_0)\|_{V} + \left\|\widehat{\phi}(\chi(t_n))-
\widehat{\phi}(\chi(t_0)) \right\|_{L^1(\Omega)} \right)\to 0\quad
\text{as $n \to +\infty$}\,.$$ Indeed, thanks to regularity , for all $T>0$, the function $\chi:[0,T] \to V$ is continuous w.r.t. the weak $V$-topology, hence $$\label{e:weakV} \chi(t_n) {\rightharpoonup}\chi(t_0) \qquad \text{in $V$.}$$ Therefore, by Lemma \[l:5.1\], we have $$\liminf_{n\to +\infty} \mathcal{E}(\chi(t_n)) \geq
\mathcal{E}(\chi(t_0))\,.$$ Combining this inequality with the continuity of the map $t \in [0,T] \mapsto
\mathcal{E}(\chi(t))$, one concludes that

\[e:converg\] $$\begin{aligned}
& \label{e:c1}
 \lim_{n \to +\infty} \int_{\Omega} |\nabla
\chi(t_n)|^2 = \int_{\Omega} |\nabla \chi(t_0)|^2,
\\
& \label{e:c2}
 \lim_{n \to +\infty}
\int_{\Omega} \widehat{\phi}(\chi(t_n))= \int_{\Omega}
\widehat{\phi}(\chi (t_0))\,.\end{aligned}$$

Clearly, , combined with , yields that $\chi(t_n) \to \chi(t_0)$ in $V$. In order to prove the additional convergence $$\label{e:stronger-conv} \|\widehat{\phi}(\chi(t_n))-
\widehat{\phi}(\chi(t_0)) \|_{L^1 (\Omega)} \to 0 \quad \text{as $n
\to +\infty$}\,,$$ we note that implies, in particular, that $$\label{e:ci-serve} \widehat{\phi}(\chi(\cdot, t_n)) \to
\widehat{\phi}(\chi(\cdot,t_0)) \qquad {\text{a.e.\ in}}\ \Omega\,.$$ In view of [@rocca-schimperna04 Lemma 4.2], , combined with and the fact that $\widehat{\phi}$ takes non-negative values, yields .

The energy identity  follows by multiplying  by $w$ (note that the latter is an admissible test function, thanks to ), by $\chi_t$, adding the resulting relations, taking into account the chain rule  and formula , and integrating in time.

It remains to prove the upper-semicontinuity with respect to the initial data. To this aim, we will exploit . Thus, let us fix a sequence of solutions $\{\chi_n \} \subset {\mathcal{S}}$ and $\chi_0 \in {X}$, with $$\label{conv-zero}
  {\operatorname{d}_X}(\chi_n(0),\chi_0) \to 0  \ \text{as $n \to +\infty$, so that, in particular, ${\mathcal{E}}(\chi_n(0)) \to
  {\mathcal{E}}(\chi_0)$.}$$ Identity  yields that there exists a constant $C>0$ such that, for all $n \in {\mathbb{N}}$, $$\label{aprio1}
  \delta\int_0^t \int_{\Omega} |\partial_t \chi_n|^2 + \int_0^t
\int_{\Omega} \alpha'(w_n)|\nabla w_n|^2 + {\mathcal{E}}(\chi_n(t)) =
{\mathcal{E}}(\chi_n(0)) \leq C \ \ \text{for all $t\geq 0$}.$$ Arguing as in Section \[s:3.1\], we obtain estimates , , , , , and  for the sequence $\{(\chi_n,w_n)\} $, on every interval $(0,T)$. Therefore, with a diagonalization procedure, we find a subsequence $\{(\chi_{n_k}, w_{n_k}) \}$ and functions $(\chi,w):(0,+\infty) \to {X}\times V$ for which –, , and hold on every interval $(0,T)$, for all $T>0$. Using all the aforementioned relations, we have $\chi(0)=\chi_0$ and, arguing as in Section \[s:3.3\], we conclude that $\chi \in {\mathcal{S}}$. In order to prove that $$\label{e:conv-chink} \text{for all $t \geq 0$}, \ \  \left( \|
\chi_{n_k}(t) -\chi(t)\|_{V} + \left\|
\widehat{\phi}(\chi_{n_k}(t))- \widehat{\phi}(\chi(t)) \right\|_{L^1
(\Omega)} \right)\to 0 \ \ \ \text{as $k \to +\infty$,}$$ we first obtain some *enhanced* convergence for the sequence $\{ w_{n_k}\}$. To this aim, we note that, for every $T>0$, there holds $$\begin{aligned}
\mathsf{c}_\alpha \limsup_{k \to +\infty} \int_0^T \int_{\Omega}
|w_{n_k}|^{2p+2} & \leq \limsup_{k \to +\infty}   \int_0^T
\int_{\Omega}  \alpha(w_{n_k}) w_{n_k} - \liminf_{k \to +\infty}
\int_0^T \int_{\Omega} \Psi(w_{n_k}) \\ & \leq  \int_0^T
\int_{\Omega} \alpha(w) w -  \int_0^T \int_{\Omega} \Psi(w)  =
\mathsf{c}_\alpha \int_0^T \int_{\Omega} |w|^{2p+2}\,.
\end{aligned}$$ Indeed, the first inequality follows from , the second one from the second convergence in , and from , together with the convexity of $\Psi$ (thanks to Ioffe’s Theorem [@ioffe77]), and the third one from  again. Taking into account the fact that $$\liminf_{k \to +\infty} \int_0^T \int_{\Omega} |w_{n_k}|^{2p+2}  \geq
\int_0^T \int_{\Omega} |w|^{2p+2},$$ due to , we have $$w_{n_k} \to w \qquad \text{in $L^{2p+2}(0,T; L^{2p+2} (\Omega))$ \ \
for all $T>0$}$$ and, thus, finally, $$\label{strong-w} w_{n_k} \to w \qquad \text{in measure in $\Omega
\times (0,T)$ for all $T>0$.}$$ As a consequence, for all $t \geq 0$, $$\label{e:ioffe-again} \liminf_{k \to +\infty} \int_0^t \int_{\Omega}
\alpha'(w_{n_k}) |\nabla w_{n_k}|^2 \geq \int_0^t \int_{\Omega}
\alpha'(w) |\nabla w|^2\,,$$ thanks to the convergence in measure , the weak convergence  for $\{ \nabla w_{n_k}\}$ in $L^2 (0,T; H)$ for all $T>0$, and again Ioffe’s Theorem \[th-ioffe\]. Hence, passing to the limit in the energy identity  (written for the functions $(\chi_{n_k},w_{n_k})$), we infer, for all $t \geq 0$, $$\label{e:elem-arg}
\begin{aligned}
 \delta\int_0^t \int_{\Omega} |\partial_t \chi|^2  & + \int_0^t
\int_{\Omega} \alpha'(w)|\nabla w|^2 + {\mathcal{E}}(\chi(t)) \\ & \leq
\liminf_{k \to +\infty} \left(  \delta\int_0^t \int_{\Omega}
|\partial_t \chi_{n_k}|^2 + \int_0^t \int_{\Omega}
\alpha'(w_{n_k})|\nabla w_{n_k}|^2 + {\mathcal{E}}(\chi_{n_k}(t))  \right)
\\ & \leq
\limsup_{k \to +\infty} \left(  \delta\int_0^t \int_{\Omega}
|\partial_t \chi_{n_k}|^2 + \int_0^t \int_{\Omega}
\alpha'(w_{n_k})|\nabla w_{n_k}|^2 + {\mathcal{E}}(\chi_{n_k}(t))  \right)
\\
&  = \lim_{k \to +\infty} {\mathcal{E}}(\chi_{n_k}(0))= {\mathcal{E}}(\chi_0) =
\delta\int_0^t \int_{\Omega} |\partial_t \chi|^2 + \int_0^t
\int_{\Omega} \alpha'(w)|\nabla w|^2 + {\mathcal{E}}(\chi(t))\,,
\end{aligned}$$ where the first inequality follows from –, , and the fact that ${\mathcal{E}}$ is lower-semicontinuous w.r.t. the $H$-topology, the third one from , the fourth one from , and the last equality from the *energy identity*  satisfied by all solutions in $\mathcal{S}$. With an elementary argument, we deduce from  that, for all $t>0$, $$\int_0^t \int_{\Omega} |\partial_t \chi_{n_k}|^2 \to \int_0^t
\int_{\Omega} |\partial_t \chi|^2, \quad \text{whence} \quad
\chi_{n_k} \to \chi \ \ \text{in $H^1(0,t;H)$,}$$ as well as $${\mathcal{E}}(\chi_{n_k}(t))  \to {\mathcal{E}}(\chi(t)).$$ Arguing in the same way as throughout – and again invoking [@rocca-schimperna04 Lemma 4.2], we obtain . This concludes the proof.

Proof of Theorem \[th:4\] {#ss:5.2}
-------------------------

#### Eventual boundedness.

In order to check that ${\mathcal{S}}$ is eventually bounded, we fix a ball $B(0,R)$ centered at $0$ of radius $R$ in ${X}$, some initial datum $\chi_0 \in B_{X}(0,R)$, namely satisfying (recall that we can assume that $\widehat{\phi}$ is a positive function) $$\label{e:in-a-ball}
   \|\chi_0\|_V+\int_{\Omega}\widehat{\phi}(\chi_0) \leq
 R,$$ and consider a generic trajectory $\chi \in {\mathcal{S}}$ starting from $\chi_0$. Recalling the energy identity , we find, for all $t \geq 0$, $$\label{e:eb1} \int_{\Omega}\widehat{\phi}(\chi(t)) \leq
{\mathcal{E}}(\chi(t)) \leq {\mathcal{E}}(\chi_0) \leq R, \qquad \int_{\Omega}|\nabla
\chi(t)|^2 \leq 2R\,.$$ Now, taking into account the fact that $m(\chi(t)) = m(\chi_0)$ for all $t \geq 0$ (cf. ), we deduce from a bound for $\| \chi \|_{L^\infty
 (0,+\infty;V)}$. Hence, there exists $R'>0$ such that ${\operatorname{d}_X}(\chi(t),0) \leq R'$ for all $t \geq
0$. Since $\chi_0$ is arbitrary, we conclude that the evolution of the ball $B_{X}(0,R)$ is contained in the ball $B_{X}(0,R')$.

#### Compactness.

In order to verify that ${\mathcal{S}}$ is compact, we consider a sequence $\{\chi_n\}\subset{\mathcal{S}}$ such that $\{\chi_n(0)\}$ is bounded in ${X}$. We write the energy identity  and, as in the proof of Proposition \[prop:2.1\], deduce that there exist a subsequence $\{(\chi_{n_k}, w_{n_k}) \}$ and functions $(\chi,w):(0,+\infty)
\to {X}\times V$ for which convergences –, , and hold on every interval $(0,T)$ for all $T>0$. However, we cannot prove that $$\label{e:conv-chink-comp}   \left( \| \chi_{n_k}(t) -\chi(t)\|_{V} +
\left\|\widehat{\phi}(\chi_{n_k}(t))- \widehat{\phi}(\chi(t))
\right\|_{L^1(\Omega)} \right)\to 0 \ \ \ \text{for all $t>0$},$$ arguing in the same way as throughout –, for, in this case, we do not have the convergence of the initial energies ${\mathcal{E}}(\chi_{n_k}(0))$ at our disposal. Then, we rely on the following procedure (see also [@rossi-segatti-stefanelli08; @segatti06] for the use of an analogous argument).

First, we apply Helly’s compactness principle (with respect to the pointwise convergence) for monotone functions to the functions $t \mapsto {\mathcal{E}}(\chi_{n_k}(t))$, which are non-increasing in view of the energy identity . Thus, up to a (not relabeled) subsequence, there exists a non-increasing function $\mathscr{E}:
[0,+\infty) \to {\mathbb{R}}$ such that $$\label{helly} \mathscr{E}(t):= \lim_{k \to +\infty}
{\mathcal{E}}(\chi_{n_k}(t)) \qquad \text{for all $t \geq 0$}.$$ By the lower-semicontinuity of ${\mathcal{E}}$ (w.r.t. the $H$-topology), we find

$$\label{helly-ineq} {\mathcal{E}}(\chi(t)) \leq \mathscr{E}(t) \qquad
\text{for all $t \geq 0$.}$$

On the other hand, ensures that, up to a further extraction, for almost all $s \in (0,t)$, $$\label{e:enhanced-conv} \chi_{n_k}(s) \to \chi(s) \ \  \text{in
$H^{2-{\varepsilon}}(\Omega)$ for all ${\varepsilon}>0$, whence} \ \ \chi_{n_k}(s) \to
\chi(s) \ \ \text{in $H^1 (\Omega) \cap L^\infty(\Omega) $.}$$ Thus, in particular, $$\label{e:pointwise} \widehat{\phi}(\chi_{n_k}(\cdot,s)) \to
\widehat{\phi}(\chi(\cdot,s)) \qquad {\text{a.e.\ in}}\ \Omega\,.$$ Moreover, for every $\mathcal{O} \subset \Omega$, there holds $$\label{e:unifinte}
\begin{aligned}
 \int_{\mathcal{O}}
|\widehat{\phi}(\chi_{n_k}(s))|  & \leq
|\mathcal{O}||\widehat{\phi}(0)| +\frac{C_{\phi,2}}2
\int_{\mathcal{O}} |\chi_{n_k}(s)|^2 + \int_{\mathcal{O}}
|{\phi}(\chi_{n_k}(s))| |\chi_{n_k}(s)| \\ &  \leq C
\left(|\mathcal{O}|+
\int_{\mathcal{O}}|{\phi}(\chi_{n_k}(s))|\right)\,,
\end{aligned}$$ where the first inequality follows from  and the second one from . Notice that the right-hand side of  tends to zero as $|\mathcal{O}| \to 0$, since the sequence $\{ {\phi}(\chi_{n_k}(s)) \}$ is uniformly integrable in $L^1 (\Omega)$ thanks to . Hence, yields that $\{ \widehat{\phi}(\chi_{n_k}(s)) \}$ is itself uniformly integrable in $L^1 (\Omega)$. Combining this with , in view of Theorem \[t:ds\] we conclude that $\widehat{\phi}(\chi_{n_k}(s)) \to \widehat{\phi}(\chi(s))$ in $L^1 (\Omega)$. Finally, we have shown that there exists a negligible set $\mathscr{N} \subset (0,+\infty)$ such that $$\label{e:convene} \mathscr{E}(s) = \lim_{k \to +\infty}
{\mathcal{E}}(\chi_{n_k}(s)) = {\mathcal{E}}(\chi(s)) \qquad {\text{for a.a.}}\, s \in
(0,+\infty) \setminus \mathscr{N}\,.$$

We are now in a position to carry out the argument for  (which bypasses the lack of convergence of the initial data in the phase space ), using the fact that the energy identity  holds for all $t>0$. Indeed, for every fixed $t>0$ and for all $s \in (0,t) \setminus \mathscr{N}$, we can pass to the limit in the energy identity , written for the sequence $(\chi_{n_k}, w_{n_k})$ on the interval $(s,t)$. Note indeed that convergences – and for $(\chi_{n_k}, w_{n_k})$ hold on $(s,t)$. Proceeding as above, we then deduce once more that $$\lim_{k \to +\infty} \int_s^t \int_{\Omega} \alpha(w_{n_k}) w_{n_k} =
\int_s^t \int_{\Omega} \alpha(w) w\,,$$ whence $w_{n_k} \to w$ in $L^{2p+2}(s,t;L^{2p+2}(\Omega))$. Therefore, repeating the very same passages as in  and relying on , we find $$\begin{aligned}
 \delta\int_s^t \int_{\Omega} |\partial_t \chi|^2  & + \int_s^t
\int_{\Omega} \alpha'(w)|\nabla w|^2 + {\mathcal{E}}(\chi(t)) \\ & = \lim_{k
\to +\infty} \left(\delta\int_s^t \int_{\Omega} |\partial_t
\chi_{n_k}|^2 + \int_s^t \int_{\Omega} \alpha'(w_{n_k})|\nabla
w_{n_k}|^2 + {\mathcal{E}}(\chi_{n_k}(t))  \right)\,,
\end{aligned}$$ which gives $$\mathscr{E}(t) = \lim_{k \to +\infty} {\mathcal{E}}(\chi_{n_k}(t)) =
{\mathcal{E}}(\chi(t)) \quad\text{for all $t>0$},$$ and, finally, .

#### Lyapunov function and rest points.

We now verify that ${\mathcal{E}}$ acts as a Lyapunov functional for ${\mathcal{S}}$. Actually, ${\mathcal{E}}$ clearly is continuous on ${X}$ and decreasing along all solutions, thanks to the energy identity . Furthermore, assume that, along some $\chi \in
{\mathcal{S}}$, the map $t \in [0,+\infty) \mapsto {\mathcal{E}}(\chi(t))$ is constant. Then, in view of , we find $\nabla w \equiv 0$ and $\chi_t \equiv 0$ a.e. in $(0,+\infty)$, so that $\chi(t) \equiv \chi(0) $ for all $t \in
[0,+\infty)$. Analogously, we immediately find that $\bar{\chi} \in
{X}$ is a rest point for ${\mathcal{S}}$ if and only if it satisfies the stationary system

\[e:sub\] $$\begin{aligned}
& \label{e:sub1} A (\alpha(\bar{w})) =0 \quad {\text{a.e.\ in}}\ \Omega\,,
\\
& \label{e:sub2} A \bar{\chi}+ \phi(\bar{\chi}) =\bar{w} \quad {\text{a.e.\ in}}\
 \Omega\,.\end{aligned}$$

#### Conclusion of the proof.

We apply Theorem \[thm:ball1\] and Remark \[rem:restriction\_to\_invariant\_set\] with the choice $\mathcal{D}:=\mathcal{D}_{m_0} $ for some $m_0>0$ (cf. ). Thanks to  (recall the second a priori estimate in Section \[s:3.1\]), for all $\chi_0 \in \mathcal{D}_{m_0} $, every solution starting from the initial datum $\chi_0$ remains in $\mathcal{D}_{m_0} $, so that the first condition in  is satisfied. To check the second one, we fix some $\bar{\chi} \in
{Z({\mathcal{S}})}$ with $|m(\bar{\chi})| \leq m_0 $. It follows from  and  that $\nabla \bar{w} \equiv
0$, so that $\bar{w}$ is constant in $\Omega$. Hence, we test  by $\bar{\chi} - m(\bar{\chi})$. Since $\bar{w}= m(\bar{w})$, we infer that $$\label{est:rest-points}
 \| \nabla \bar{\chi} \|_H^2 + \int_{\Omega} \phi(\bar{\chi})
 (\bar{\chi}-m(\bar{\chi})) \leq 0\,.$$ On the other hand, ensures that estimate  holds, so that there exist constants $\mathcal{K}_{m_0}, \,\mathcal{K}_{m_0}^1>0$, only depending on $m_0$, such that $$\label{est-aggiunta} \int_{\Omega}|\phi(\bar{\chi})| \leq
\mathcal{K}_{m_0} \int_{\Omega} \phi(\bar{\chi})
 (\bar{\chi}-m(\bar{\chi})) + \mathcal{K}_{m_0}^1\,.$$ Collecting  and , we deduce that $$\| \nabla \bar{\chi} \|_H^2 +
\frac1{\mathcal{K}_{m_0}}\int_{\Omega}|\phi(\bar{\chi})|  \leq
\frac{\mathcal{K}_{m_0}^1}{\mathcal{K}_{m_0}}\,,$$ whence, in particular, $$|m(\bar{w})|=|m(\phi(\bar{\chi}))| \leq
\frac{\mathcal{K}_{m_0}^1}{|\Omega|}.$$ Taking into account the fact that $\nabla \bar{w}=0$ (so that $\bar{w}$ is a constant) and that $|m(\bar{\chi})| \leq m_0 $, we conclude that $$\label{e:est-rest} \exists\,\mathcal{K}_{m_0}^{2}>0\,: \ \ \forall\,
\bar{\chi} \in {Z({\mathcal{S}})}\cap\mathcal{D}_{m_0}\ \ \| \bar{\chi} \|_{V} +
| \bar{w} | \leq \mathcal{K}_{m_0}^2\,.$$ Thus, a comparison in  and the standard elliptic regularity estimate (cf. also the calculations developed throughout –), yield $$\label{e:later}
 \exists\,\mathcal{K}_{m_0}^{3}>0\,:  \ \ \forall\,
\bar{\chi} \in {Z({\mathcal{S}})}\cap\mathcal{D}_{m_0} \ \ \| {\phi}(\bar{\chi})
\|_{L^6 (\Omega)}+ \| \bar{\chi}\|_{W^{2,6}(\Omega)} \leq
\mathcal{K}_{m_0}^3\,,$$ whence, in particular, an estimate for $\bar{\chi}$ in $L^\infty
(\Omega)$. Then, using , we readily infer that $$\label{e:est-rest2} \exists\,\mathcal{K}_{m_0}^{4}>0\,: \ \
\forall\, \bar{\chi} \in {Z({\mathcal{S}})}\cap\mathcal{D}_{m_0} \ \ \|
\widehat{\phi}(\bar{\chi}) \|_{L^6 (\Omega)}  \leq
\mathcal{K}_{m_0}^4\,.$$ Finally, and yield that ${Z({\mathcal{S}})}\cap\mathcal{D}_{m_0} $ is bounded in the phase space ${X}$, and the existence of the global attractor follows from Theorem \[thm:ball1\].

In fact, with the same calculations as in the above lines, joint with a boot-strap argument, one easily proves that $$\label{rest}
 \forall\, p \in [1,+\infty)\,: \ \ \exists\,C_p>0 \ \
\bar{\chi} \in {Z({\mathcal{S}})}\cap\mathcal{D}_{m_0} \quad \|
\bar{\chi}\|_{W^{2,p}(\Omega)} + \|
\widehat{\phi}(\bar{\chi})\|_{L^p(\Omega)} \leq C_p.$$ Then, estimate  is a straightforward consequence of  and .

\[s:6\]

Proof of Theorems \[th:3\] and \[th:3.2\] {#ss:6.1}
=========================================

#### Proof of Theorem \[th:3\].

Within this proof, we denote by $c_\delta$ a positive constant depending on $\delta>0$ and on quantities . Referring to the notation of the statement of Theorem \[th:3\], let us set ${\underline{\chi}_{0}}:=
\chi_{0}^{1}-\chi_{0}^{2}$, $ {\underline{\chi}}:=\chi_1 -\chi_2 $, and ${\underline{w}}:=
w_1-w_2$. The pair $ ({\underline{\chi}},{\underline{w}}) $ obviously satisfies $$\begin{aligned}
  &&
   {\underline{\chi}}_t  + A(\alpha (w_1))-  A(\alpha (w_2)) =0
 \quad  {\text{a.e.\ in}}\ \Omega \times (0,T),
\label{unodifb}\\
&&
 \label{duedifb}
\delta {\underline{\chi}}_t +A {\underline{\chi}}+ \chi_{1}^3-\chi_{2}^3  -{\underline{\chi}}= {\underline{w}}\quad
{\text{a.e.\ in}}\ \Omega \times (0,T).\end{aligned}$$ Following the proof of [@rossi05 Prop. 2.1], we test by $ \mathcal{N}\left({\underline{w}}- {m}({\underline{w}}) \right)$, by $
\mathcal{N}( {\underline{\chi}}_t)+ {\underline{\chi}}$, add the resulting equations, and integrate over $(0,t)$, $ t \in (0,T)$. We refer to the proof of [@rossi05 Prop. 2.1] for all the detailed computations, leading to (cf. [@rossi05 (3.51)]) $$\label{basic-cont-dep}
 \begin{aligned}
\int_0^t \| {\underline{w}}\|_{H}^2    + \delta \int_0^t \|{\mathcal{N}}({\underline{\chi}}_t)
{\|_{V}}^2+ \delta \|{\underline{\chi}}(t)  \|_{H}^2
 + \int_0^t \|
\nabla {\underline{\chi}}\|_{H}^2
   \leq C\left(\|{\underline{\chi}_{0}}\|_H^2  + \int_0^t \|{\underline{\chi}}\|_H^{2} \right).
\end{aligned}$$ An easy application of Gronwall’s lemma to the function $t \mapsto
\|{\underline{\chi}}(t)\|_H^{2} $ entails $$\label{eq:stima-cont-dep-h} \| {\underline{\chi}}\|_{C^0 ([0,t];H) \cap L^2
(0,t;V)}+ \| {\underline{\chi}}_t\|_{L^2 (0,t;V')}+
 \| {\underline{w}}\|_{L^2 (0,t;H)}\leq c_\delta  \| {\underline{\chi}_{0}}\|_H.$$ Furthermore, exploiting  and the above , it follows from the Hölder inequality that $$\label{b}
\begin{aligned}
&\| \phi(\chi_1) -\phi(\chi_2) \|_{L^2 (0,t;H)}^2 \\ & \leq C
\int_0^t \int_\Omega |{\underline{\chi}}|^2 \left(\chi_1^2 +\chi_2^2  +1\right)^2
\\
&\leq C \int_0^t \left(\| \chi_1  \|_{L^6 (\Omega)}^4 + \| \chi_2
\|_{L^6 (\Omega)}^4 \right) \| {\underline{\chi}}\|_{L^6 (\Omega)}^2 + C
\int_0^t \int_\Omega |{\underline{\chi}}|^2
\\
&\leq C \left(\| \chi_1 \|_{L^{\infty}(0,T;L^{6}(\Omega))}^4 + \|
\chi_2 \|_{L^{\infty}(0,T;L^{6}(\Omega))}^4 +1\right) \| {\underline{\chi}}\|_{L^2 (0,t;V)}^2 \leq c_\delta \| {\underline{\chi}_{0}}\|_H^2.
\end{aligned}$$ Next, we test  by ${\underline{\chi}}_t $ and integrate in time to obtain $$\label{e:6.5}
\begin{aligned}
\frac\delta2\int_0^t \| {\underline{\chi}}_t  \|_H^2  + \frac12 \|
\nabla({\underline{\chi}}(t)) \|_H^2 \leq \frac12 \|\nabla{\underline{\chi}_{0}}\|_{H}^2
+c_\delta \left(\int_0^t \| {\underline{w}}\|_{H}^2 +\int_0^t \|\phi(\chi_1)
-\phi(\chi_2)\|_H^2 + \int_0^t \| {\underline{\chi}}\|_{H}^2 \right)\,.
\end{aligned}$$ In view of –, we readily infer the continuous dependence estimate  for ${\underline{\chi}}$ in $C^0 ([0,t];V) \cap H^1 (0,t;H)$. Then, the estimate for $\| {\underline{\chi}}\|_{L^2 (0,t;Z)}$ follows from – by a comparison argument.

#### Proof of Theorem \[th:3.2\].

Referring to the notation of the proof of Theorem \[th:3\], we again test by $ \mathcal{N}\left({\underline{w}}- {m}({\underline{w}}) \right)$, by $
\mathcal{N}( {\underline{\chi}}_t)+ {\underline{\chi}}$, add the resulting equations, and integrate over $(0,t)$, $ t \in (0,T)$. Developing the same calculations as in the above lines, we note that the chain of inequalities is now trivial, since under the present assumptions the functions $\chi_1$ and $\chi_2$ are estimated in $L^\infty (0,T;Z)$ (see Proposition \[prop:regularized\]). On the other hand, the following term: $$I :=\int^t_0 \int_\Omega m({\underline{w}})\left(\alpha(w_1) -\alpha(w_2)\right),$$ which was easily estimated in the proof of Theorem \[th:3\], now needs to be carefully handled because of the (at most) quadratic controlled growth of $\alpha^\prime$. Indeed, observe that $$\begin{aligned}
\vert I\vert &\le \int_0^t \Vert m({\underline{w}})\Vert_{L^\infty(\Omega)}
\Vert\alpha(w_1) -\alpha(w_2)\Vert_{L^1(\Omega)}
\\
&\le C\int_0^t \left(\Vert m({\underline{w}})\Vert_{L^1(\Omega)}\int_\Omega
(1+\vert w_1\vert^{2p} +\vert w_2\vert^{2p}){\underline{w}}\right)
\\
&\le C \int_0^t \left(\Vert \phi(\chi_1)
-\phi(\chi_2)\Vert_{L^1(\Omega)} (1+\Vert
w_1\Vert^{2p}_{L^\infty(\Omega)} +\Vert
w_2\Vert^{2p}_{L^\infty(\Omega)})\Vert {\underline{w}}\Vert_{L^1(\Omega)}
\right)
\\
&\le C \int_0^t \left(\Vert{\underline{\chi}}\Vert_{L^1(\Omega)} (1+\Vert
w_1\Vert^{2p}_{L^\infty(\Omega)} +\Vert
w_2\Vert^{2p}_{L^\infty(\Omega)})\Vert {\underline{w}}\Vert_{L^1(\Omega)}
\right)
\\
&\le \varrho \int_0^t\Vert {\underline{w}}\Vert^2_{H} + C_\varrho \int_0^t
(1+\Vert w_1\Vert^{4p}_{L^\infty(\Omega)} +\Vert
w_2\Vert^{4p}_{L^\infty(\Omega)}) \Vert{\underline{\chi}}\Vert^2_{H}
\end{aligned}$$ for some $\varrho \in (0,1)$ and $C_{\varrho}>0$. This modification gives, in place of , $$\begin{aligned}
&(1-\varrho)\int_0^t \| {\underline{w}}\|_{H}^2    + \delta \int_0^t \|{\mathcal{N}}({\underline{\chi}}_t) {\|_{V}}^2+ \delta \|{\underline{\chi}}(t)  \|_{H}^2
 + \int_0^t \|
\nabla {\underline{\chi}}\|_{H}^2\\
   &\leq C\left(\|{\underline{\chi}_{0}}\|_H^2  +
   \int_0^t (1+\Vert w_1\Vert^{4p}_{L^\infty(\Omega)}
+\Vert w_2\Vert^{4p}_{L^\infty(\Omega)}) \Vert{\underline{\chi}}\Vert^2_{H}\right).
\end{aligned}$$ Thus, recalling , we can use Gronwall’s lemma to deduce . Estimate can be obtained by arguing as in the proof of Theorem \[th:3\], hence the result.

Appendix
========

We propose the following approximate system for *both* Problem \[p:1\] and Problem \[p:2\]: $$\begin{aligned}
  & \label{1-var-better-mudelta}
   \chi_t + A (\alpha_M(w)) =0 \qquad {\text{a.e.\ in}}\  \Omega \times (0,T)\,,\\
  &  \label{2-var-better-mudelta}
   \delta \chi_t+ A\chi + \phi_{{\mu}}(\chi) =w \qquad {\text{a.e.\ in}}\ \Omega \times (0,T)\,,\end{aligned}$$ depending on the parameters $\delta,\, M, \, {\mu}>0$, where $$\label{approalfa}
  \alpha_M(r)=\left\{\begin{array}{lll}
   \alpha(-M) + C_1(r-M) &{\quad\text}{if }\,r<- M,
  \\
     \alpha(r)& {\quad\text}{if }\,|r|\le M,\\
     \alpha(M) + C_1(r-M) &{\quad\text}{if }\,r> M,
   \end{array}
   \right.$$ $C_1$ being the same constant as in , and $$\label{approphi}
  \phi_{{\mu}}(r)=
\left\{
  \begin{array}{lll}
       \phi(r) &{\quad\text}{if }\,|\phi(r)|\le\frac1{{\mu}},\\
     \frac1{{\mu}}\operatorname{sign}(r) &{\quad\text}{otherwise}.
   \end{array}
   \right.$$ It is immediate to check that, for any choice of the approximation parameters $M$ and ${\mu}$, the functions $\alpha_M$ and $\phi_{{\mu}}$ are Lipschitz continuous on ${\mathbb{R}}$ and that $$\label{e:uniformly-cpt}
\begin{aligned}
&
 \alpha_M \to \alpha \qquad \text{uniformly
on compact subsets of ${\mathbb{R}}$ as $M \nearrow +\infty$,}
\\
&
 \phi_\mu \to \phi \qquad \text{uniformly
on compact subsets of ${\text{dom}}(\phi)$ as $\mu \searrow 0$.}
\end{aligned}$$ Of course, the Lipschitz constants of $\alpha_M$ and $\phi_\mu$ explode as $M \nearrow +\infty$ and $\mu \searrow 0$, respectively. Let us also point out that, by construction, $$\label{e:coercivity} \alpha_{M}'(r) \geq C_1>0 \qquad \text{for all
$r \in {\mathbb{R}}$, $M
>0,$}$$ which yields that the inverse $\rho_M : {\mathbb{R}}\to {\mathbb{R}}$ of $\alpha_M$ is Lipschitz continuous, with $$\label{e:impo-conse} |\rho_M(x) -\rho_M(y)| \leq \frac{1}{C_1}|x-y|
\quad \text{for all $x,y \in {\mathbb{R}}$, $M
>0.$}$$ What is more, relying on convergence of $\phi_{\mu}$ to $\phi$, one can also check that, for $\mu>0$ sufficiently small (say $0<\mu\leq \mu_*$), and hold on this approximate level as well, i.e., $$\label{e:2.2.14-approx}
\begin{aligned}
 \forall\, m \in {\text{dom}}(\phi)=(a,b)\ \  \exists\, C_m,\
C_m'>0 \, : \ \ &\forall\, 0<\mu\leq \mu_* \quad  \forall\, r \in
(a-m,b-m) \\ & |\phi_{\mu}(r+m)| \leq C_m \phi_{\mu}(r+m)r +C_m'\,,
\end{aligned}$$ as well as $$\label{hyp:3-appr} \ \ \exists\, C>0\, : \ \ \forall\, 0<\mu\leq
\mu_* \quad  \quad \forall\, r \in (a,b) \quad
|\phi_{\mu}(r)|^{\sigma} \leq C \left( \widehat{\phi_{\mu}}(r)
+1\right)\,,$$ with $\sigma \in (0,1)$ as in , in particular, complying with the compatibility condition .

It was proved in [@rossi05 Thm. 2.1] that, for every $\delta,\,  M, \, {\mu}>0$, there exists a unique pair $({\chi},{w})$, with $$\label{initial-regularity}
 \begin{aligned}
  & {\chi}\in L^2 (0,T;Z) \cap L^\infty (0,T;V) \cap H^1 (0,T;H),\\
  &{w}\in L^2 (0,T;V),
 \end{aligned}$$ solving the Cauchy problem for system –, supplemented with some initial datum $\chi^0 \in V$.

#### Problem $\mathbf{P}_{\delta,{\mu}}$.

In what follows, we approximate the initial datum $\chi_0 \in V$ in  by a sequence $$\label{basta}
 \{\chi_{0,\mu}\} \subset H^4(\Omega) \quad \text{with} \quad \chi_{0,\mu}
  {\rightharpoonup}\chi_0 \ \ \text{in $V$} \ \ \text{and} \ \ \sup_{\mu>0}\|
  \widehat{\phi}_{{\mu}}(\chi_{0,\mu})\|_{L^1(\Omega)}<+\infty$$ (for example, we can construct $\{ \chi_{0,\mu}\}$ by applying (twice) the elliptic regularization procedure developed in the proof of [@bcgg Prop. 2.6]).

For every $\delta,\,  M, \, {\mu}>0$, we call $\mathbf{P}_{\delta,M,{\mu}}$ the initial and boundary value problem obtained by supplementing the PDE system – with the initial condition $$\label{app-init}
  \chi(0)=\chi_{0,\mu} \quad \text{in $H^4(\Omega)$}.$$ In the following Section \[ss:a.1\], we will prove some further regularity of the approximate solutions. In this way, we will justify, on the level of the approximate Problem $\mathbf{P}_{\delta,M,{\mu}}$, the estimates formally performed in Section \[s:3.1\]. Hence, in Section \[ss:a.2\], we will develop the rigorous proof of Theorem \[th:1\] by relying on the aforementioned estimates and by passing to the limit in Problem $\mathbf{P}_{\delta,M, {\mu}}$, first as $\delta\searrow 0$ for $M,\,{\mu}>0$ fixed, then as $M \nearrow +\infty$ for ${\mu}>0$ fixed, and, finally, as ${\mu}\searrow 0$.

Furthermore, it would be possible to give a rigorous proof of Theorem \[th:2\] by passing to the limit in Problem $\mathbf{P}_{\delta,M,{\mu}}$ first as $M \nearrow +\infty$ for ${\mu}>0$ fixed, and then as ${\mu}\searrow 0$. However, we are not going to enter into the details of the latter procedure, which follows the very same lines as the one for Theorem \[th:1\].

In what follows, we denote by $C_{\delta,M,{\mu}}$ various constants (which can differ from occurrence to occurrence, even in the same line), depending on the parameters $\delta$, $M$, and ${\mu}$, and such that $C_{\delta,M,{\mu}} \nearrow
+\infty$ as either $\delta \searrow 0$, or $M \nearrow +\infty$, or ${\mu}\searrow 0$. The symbols $C_{\delta,\mu}$, $C_{M,\mu}$, and $C_{\mu}$ have an analogous meaning.

Enhanced regularity estimates on the approximate problem {#ss:a.1}
--------------------------------------------------------

#### First estimate.

We note that ${w}\in L^2 (0,T;V)$ and that, since $\phi_{\mu}$ is a Lipschitz continuous function, ${\chi}\in L^\infty(0,T;V)$ (cf. ) implies $\phi_{\mu}({\chi})  \in
L^\infty(0,T;V)$. Thus, by comparison in , we have $\delta \partial_t {\chi}+
 A{\chi}\in L^2 (0,T;V)$. Hence, testing by $A(\partial_t {\chi})$ and using the fact that ${\chi}(0)= \chi_{0,{\mu}} \in Z$, we deduce the estimate $$\label{f1}
  \|\nabla \partial_t {\chi}\|_{L^2 (0,T;H)}
   + \| A {\chi}\|_{L^\infty (0,T;H)}
  \leq C_{\delta,{\mu}},$$ whence $$\label{f1-bis}
  {\chi}\in L^\infty (0,T;Z) \cap H^1 (0,T;V).$$

#### Second estimate.

Since $\alpha_M$ is Lipschitz continuous and $w \in L^2(0,T;V)$, we have $\alpha_M({w}) \in
L^2(0,T;V)$. Estimate  and a comparison in  yield a bound for $A(\alpha_M({w}))$ in $L^2 (0,T;V)$, whence $$\alpha_M(w) \in L^2(0,T;H^3(\Omega)) \subset L^2(0,T;W^{1,\infty}(\Omega)).$$ Recalling and using the fact that $w=\rho_M (\alpha_M(w))$, we readily deduce the estimate $$\label{regosig4}
  \|{w}\|_{L^2(0,T;W^{1,\infty}(\Omega))} \leq C_{\delta,M,{\mu}}.$$

#### Third estimate.

Using a parabolic regularity argument in and relying on regularity for the approximate initial datum $\chi_{0,{\mu}}$, we deduce that $$\label{cf-later}
  \| \partial_t {\chi}\|_{L^2 (0,T;W^{1,3+\epsilon}(\Omega))} + \| A
  {\chi}\|_{L^2 (0,T;W^{1,3+\epsilon}(\Omega))} \leq C_{\delta,M,\mu},$$ where $\epsilon>0$ is a suitable number. More precisely, since $\chi_{0,\mu}\in H^4(\Omega)\subset W^{3,6}(\Omega)$, the above formula holds for any $\epsilon\in(0,3]$ (cf. inequality below for a justification). Thus, by interpolation, we obtain that $\nabla {\chi}$ belongs to $ H^{1/2}
(0,T; W^{1,3+\epsilon}(\Omega))$ and, recalling the continuous embedding $W^{1,3+\epsilon}(\Omega) \subset L^\infty (\Omega)$, we conclude that $$\label{regosig5}
  \| \nabla{\chi}\|_{L^\infty (0,T;L^\infty(\Omega))} \leq
   C_{\delta,M,{\mu}}.$$

#### Fourth estimate.

Notice that, for almost all $t \in (0,T)$, the function $\nabla(|{w}(t)|^p {w}(t))= (p+1)|{w}(t)|^p  \nabla {w}(t)$ belongs to $L^2(\Omega)$, thanks to  and . Hence, for a.a. $t \in (0,T)$, we can test  by $|{w}(t)|^p {w}(t)$, which yields $$\label{f3}
\begin{aligned}
\int_{\Omega} &  |{w}(t)|^{p+2} \\ & = \int_{\Omega} \nabla
{\chi}(t) \cdot \nabla(|{w}(t)|^p {w}(t)) + \int_{\Omega}
\phi_{\mu}({\chi}(t)) |{w}(t)|^p {w}(t) + \delta \int_{\Omega}
\partial_t {\chi}(t) |{w}(t)|^p {w}(t)
\\ &
\begin{aligned}
= (p+1)\int_{\Omega}  |{w}(t)|^p \nabla {\chi}(t) \cdot  \nabla
{w}(t)  &  + \int_{\Omega} \phi_{\mu}({\chi}(t)) |{w}(t)|^p
{w}(t) \\ & - \delta(p+1) \int_{\Omega}
\alpha_M'({w}(t))|{w}(t)|^p | \nabla {w}(t)|^2,
\end{aligned}
\end{aligned}$$ the second equality following from equation . We estimate the second term on the right-hand side of the above equality by using the bound for $\phi_{\mu}({\chi})$ in $L^\infty (0,T; L^\infty (\Omega))$, due to  and the Lipschitz continuity of $\phi_{\mu}$. We deal with the first integral term as follows: $$\label{e:f2}
\begin{aligned}
  \left|\int_{\Omega}  |{w}(t)|^p \nabla {\chi}(t) \cdot  \nabla {w}(t)  \right|
   & \le \big\| |{w}(t)|^{p/2}\nabla {w}(t)\big\|_{L^2(\Omega)}
    \big\| |{w}(t)|^{p/2} \big\|_{L^2(\Omega)} \big\|\nabla {\chi}(t)\big\|_{L^\infty}
  \\ &  \le \varrho {\int_\Omega}|w(t)|^p |\nabla w(t)|^2
    + C_{\delta,M,\mu} \int_{\Omega}| w(t) |^p
    \end{aligned}$$ for some suitable positive constant $\varrho$, where we have also used . Now, recalling , we estimate the last summand on the right-hand side of  by $$-\delta(p+1) \int_{\Omega} \alpha_M'({w}(t))|{w}(t)|^p | \nabla
{w}(t)|^2 \leq -\delta(p+1)C_1 \int_{\Omega} |{w}(t)|^p |\nabla
{w}(t)|^2,$$ and we move the above term to the left-hand side of . Then, we combine the latter inequality with , in which we choose $\varrho=\frac{\delta (p+1)C_1}{4}$. We thus obtain, for a.a. $t \in (0,T)$, $$\label{co31}
\int_{\Omega} |{w}(t)|^{p+2}
   +\frac34\delta(p+1)C_1 \int_{\Omega} |{w}(t)|^p |\nabla {w}(t)|^2
   \le C_{\delta,M,{\mu}} \left(\int_{\Omega}|{w}(t)|^{p+1} + \int_{\Omega}| w(t)
   |^p\right)\,.$$ Thus, we finally infer that $$\label{regosig6}
  w \in L^\infty(0,T;L^p(\Omega)) \quad \text{for all $p\in[1,\infty)$,}$$ whence, by the Lipschitz continuity of $\alpha_M$, $$\label{regosig7}
  \alpha_M(w) \in L^\infty(0,T;L^p(\Omega)) \quad \text{for all $p\in[1,+\infty)$.}$$

Rigorous proof of Theorem \[th:1\] {#ss:a.2}
----------------------------------

Within this section, for all $\delta,\,{\mu}>0$, we will denote by $\{({\chi_{\delta,M,{\mu}}},{w_{\delta,M,{\mu}}})\}_{\delta,M,{\mu}}$ the family of solutions to Problem $\mathbf{P}_{\delta,M,{\mu}}$.

. For fixed ${\mu},M>0 $, we pass to the limit in Problem $\mathbf{P}_{\delta,{\mu}}$ as $\delta\searrow
0$. We then perform the same calculations as in Section \[s:3.1\] (cf. –, , –). Also relying on –, we conclude that $$\label{ff1}
 \begin{aligned}
 & \exists\, C>0\,:  \ \ \forall\, \delta,\, M, \, \mu>0 \quad
  \|{\chi_{\delta,M,{\mu}}}\|_{L^\infty (0,T;V)} + \| {w_{\delta,M,{\mu}}}\|_{L^2 (0,T;V)}
   + \| \widehat{\phi}_{\mu}({\chi_{\delta,M,{\mu}}})\|_{L^\infty (0,T;L^1 (\Omega))} \\
   & \mbox{}~~~~~~~~~~
   + \delta^{1/2} \| \partial_t {\chi_{\delta,M,{\mu}}}\|_{L^2 (0,T;L^2(\Omega))}
   + \| (\alpha_M'({w_{\delta,M,{\mu}}}))^{1/2} \nabla {w_{\delta,M,{\mu}}}\|_{L^2 (0,T;L^2(\Omega))}
   \leq C.
 \end{aligned}$$ Recalling the definition of $\phi_{\mu}$ and its Lipschitz continuity, we also have $$\label{ff1-bis}
 \begin{aligned}
  \exists\, C_\mu>0\,:  \ \ \forall\, \delta,\, M>0\quad
  \| \phi_{\mu}({\chi_{\delta,M,{\mu}}})\|_{L^\infty (0,T;V)\cap L^\infty (0,T;L^\infty
  (\Omega))} \leq C_\mu.
 \end{aligned}$$ In the same way, estimate for ${w_{\delta,M,{\mu}}}$ and the Lipschitz continuity of $\alpha_M$ yield $$\label{est-intermediate}
  \exists\, C_{M}>0\,:  \ \ \forall\, \delta,\, {\mu}>0 \quad
  \|\alpha_M({w_{\delta,M,{\mu}}})\|_{L^2 (0,T;V)} \leq C_{M}.$$ Next, a comparison in and the maximal parabolic regularity result from [@hieber-pruss] yield $$\label{hieber-pruss} c(\delta)\int_0^T \| \partial_t {\chi_{\delta,M,{\mu}}}\|_{L^6
(\Omega)}^2 + \int_0^T \| A {\chi_{\delta,M,{\mu}}}\|_{L^6 (\Omega)}^2  \leq C
\int_0^T \| \ell_{\delta,M,{\mu}}\|_{L^6 (\Omega)}^2,$$ for some $c(\delta)$ such that $c(\delta)\to 0$ as $\delta \to 0$, where we have set $$\ell_{\delta,M,{\mu}} = {w_{\delta,M,{\mu}}}-\phi_{\mu}({\chi_{\delta,M,{\mu}}}) -A\chi_{0,{\mu}}.$$ In view of estimates for ${w_{\delta,M,{\mu}}}$, for $\phi_{\mu}({\chi_{\delta,M,{\mu}}})$ in $L^\infty (0,T;V)$, and for $\{\chi_{0,{\mu}}\}$, we conclude that $$\|  \ell_{\delta,M,{\mu}} \|_{L^2 (0,T;L^6(\Omega))} \leq C_\mu.$$ Therefore, gives $$\label{inghippo}
  \exists\, C_\mu>0\,: \ \ \forall\, \delta,\, M>0 \quad \|{\chi_{\delta,M,{\mu}}}\|_{L^2
  (0,T;W^{2,6}(\Omega))} \leq C_{\mu}.$$ On the other hand, estimate and a comparison in  imply $$\label{est-chit} \exists\, C_{M}>0\,:  \ \ \forall\, \delta, \, {\mu}>0\quad
    \|\partial_t {\chi_{\delta,M,{\mu}}}\|_{L^2 (0,T;V')} \leq C_{M}.$$ On behalf of the above estimates and arguing in the very same way as in Section \[s:3.2\], we see that, for every fixed $M>0$ and $\mu>0$, there exist a sequence $\delta_k \searrow 0$ (for notational simplicity, we do not highlight its dependence on the parameters $M$ and $\mu$) and functions $(\chi_{M,{\mu}},w_{M,{\mu}},
\bar{\alpha}_{M,{\mu}})$ such that the sequence $\{(\chi_{\delta_k,M,\mu},w_{\delta_k,M,\mu}) \}$ converges to $(\chi_{M,{\mu}},w_{M,{\mu}})$, as $k \to +\infty$, in the sense specified by –, , as well as $$\begin{aligned}
  & \partial_t \chi_{\delta_k,M,\mu} {\rightharpoonup}\partial_t \chi_{M,{\mu}} \quad
   \text{in $L^2 (0,T;V')$,}\\
   & \delta_k^{1/2} \partial_t \chi_{\delta_k,M,\mu} {\rightharpoonup}0 \quad
   \text{in $L^2(0,T;H)$,}\\
   & \alpha_M(w_{\delta_k,M,\mu}) {\rightharpoonup}\bar{\alpha}_{M,{\mu}} \quad
  \text{in $L^2 (0,T;V)$.}
\end{aligned}$$ Next, arguing similarly to the (formal) proof of Theorem \[th:1\], we conclude that $$\label{post}
  \phi_{\mu}(\chi_{\delta_k,M,\mu}) \to \phi_{\mu}(\chi_{M,{\mu}}) \quad
   \text{in $L^2 (0,T;H)$.}$$ Finally, we use in the very same way as in Section \[s:3.2\] to infer $$\bar{\alpha}_{M,{\mu}} =\alpha_M (w_{M,{\mu}})$$ and $$\alpha_M(w_{\delta_k,M,\mu}) \to   \alpha_M (w_{M,{\mu}}) \quad
   \text{in $L^2 (0,T;H)$.}$$ Therefore, we conclude that the pair $(\chi_{M,{\mu}},w_{M,{\mu}})$ is a solution to the PDE system $$\begin{aligned}
  & \label{1-var-better-mu}
   \chi_t + A (\alpha_M(w)) =0 \qquad {\text{a.e.\ in}}\  \Omega \times (0,T)\,,\\
 &
 \label{2-var-better-mu} A\chi + \phi_{{\mu}}(\chi)
  = w \qquad {\text{a.e.\ in}}\ \Omega \times (0,T)\,,\end{aligned}$$ supplemented with the initial condition .\
. We now take the limit $M\nearrow +\infty$ in (the Cauchy problem for) –. Estimates  and hold for the sequence of solutions $\{(\chi_{M,{\mu}},w_{M,{\mu}})\}_M$ as well. Furthermore, using a lower-semicontinuity argument, we also deduce from that $$\label{bor} \|\chi_{M,{\mu}}\|_{L^2 (0,T;W^{2,6}(\Omega))}\leq
C_\mu\quad \text{for all $M>0.$}$$

Now, we point out that entails $$\label{stima_g} \int_0^T \int_{\Omega} \alpha_M'(w_{M,{\mu}})|\nabla
w_{M,{\mu}} |^2 \leq C \quad \text{for all $M>0$}.$$ Let us denote by $\mathcal{T}_M$ the truncation operator at level $M$ and define $$\label{def-taum} \tau_M:=\mathcal{T}_M(w_{M,{\mu}} ):= \left\{
\begin{array}{lll}  -M  &\text{if } w_{M,{\mu}} <-M,
\\
  w_{M,{\mu}} &\text{if } |w_{M,{\mu}}| \leq M,
 \\
  M & \text{if } w_{M,{\mu}} > M,
\end{array}
\right. \qquad \text{a.e.}\ t\in \Omega \times (0,T)$$ (to simplify, we omit the index $\mu$ in the notation for $\tau_M$). For later use, we also introduce ${\text{for a.a.}}\ t \in (0,T)$ the sets $$\label{def-om} \begin{cases} \mathcal{A}_M:= \left\{(x,t)\in \Omega
\times (0,T)\, : \ |w_{M,{\mu}}(x,t)| \leq M \right\},
\\
 \mathcal{O}_M:= \left\{(x,t)\in \Omega \times (0,T)\, : \
|w_{M,{\mu}}(x,t)|> M \right\},
\\
 \mathcal{O}_M^t:= \left\{x\in \Omega\, : \ (x,t) \in \mathcal{O}_M
\right\}.
\end{cases}$$ From , we also infer $$\int_0^T \int_{\Omega} \alpha'(\tau_M)|\nabla \tau_M |^2 \leq C
\quad \text{for all $M>0$},$$ whence, in view of , $$\label{stima_h} \| |\tau_M|^p\, \nabla \tau_M \|_{L^2 (0,T;H)} \leq
C
 \quad \text{for all $M>0$}.$$ Now, in order to reproduce estimates – in the present approximate setting, we test by $|\tau_{M}(t)|^p
\tau_{M}(t)$ for a.e. $t \in (0,T)$. Clearly, $$\int_{\Omega} w_{M,{\mu}}(t)|\tau_{M}(t)|^p \tau_{M}(t)
\ge\int_{\Omega} | \tau_{M}(t)|^{p+2},$$ so that we have (cf. also ) $$\label{stima_j}
 \begin{aligned}
  \int_{\Omega}   | \tau_{M}(t)|^{p+2}
   & \le (p+1)\int_{\Omega} | \tau_{M}(t)|^p \,\nabla  \chi_{M}(t) \cdot  \nabla \tau_{M}(t)
   + \int_{\Omega} \phi_{\mu}(\chi_{M,{\mu}}(t))|\tau_{M}(t)|^p \tau_{M}(t)\\
   & \leq (p+1) \| | \tau_{M}(t) |^p \,\nabla \tau_{M}(t) \|_H
     \| \nabla \chi_{M}(t)\|_H
      + C_\mu \int_{\Omega} | \tau_M(t)|^{p+1} \\
   & \leq C \||\tau_{M}(t)|^p \,\nabla \tau_{M}(t) \|_{H}
   + \frac12 \int_{\Omega} | \tau_M (t)|^{p+2}
   + C_\mu,
 \end{aligned}$$ where the second inequality follows from estimate and the last one from and Young’s inequality. Therefore, combining and , we find an estimate for $\|
\tau_M\|_{L^{p+2}(\Omega)}^{p+2}$ in $L^2 (0,T)$ with some constant $C_\mu>0$ which is independent of $M>0$. On behalf of –, from the latter bound, we infer (recall that $|\cdot|$ also denotes the Lebesgue measure) $$\label{stima_m}
\begin{aligned}
C_\mu \geq \int_{0}^T \left( \int_{\mathcal{O}_M^t} |M|^{p+2}\, {\mathrm{d}}x\right)^2 \, {\mathrm{d}}t = M^{2p+4} \int_0^T |\mathcal{O}_M^t|^2 \, {\mathrm{d}}t
\geq \frac{ M^{2p+4}}{T}|\mathcal{O}_M|^2  \quad \text{for all
$M>0$},
\end{aligned}$$ where the last inequality is a direct consequence of Jensen’s inequality. Next, we apply the nonlinear Poincaré inequality to $|\tau_M|^p \tau_M$, thus obtaining (cf. ) $$\||\tau_M|^p \tau_M  \|_{V} \leq K \left( \| \nabla
(|\tau_M|^{p} \tau_M) \|_{H} + \left|
{m}(\tau_M)\right|^{p+1}\right)\,.$$ In view of and of the definition of $\tau_M$, we find an estimate for $|\tau_M|^p \tau_M $ in $L^2 (0,T;V)$, again with some constant $C_{\mu}$ which is independent of $M>0$. Hence, using the fact that $V \subset L^6 (\Omega)$ and the growth condition for $\alpha$, we conclude that $$\label{stima_l} \| \alpha(\tau_M)\|_{L^{\rho_p}(0,T; L^{\kappa_p}
(\Omega))} \leq C_\mu  \quad \text{for all $M>0$}$$ (where the indexes $\rho_p$ and $\kappa_p$ are as in : in particular, $1<\rho_p<2$). Therefore, we have $$\begin{aligned}
\int_0^T \int_{\Omega} |\alpha_M (w_{M,{\mu}})|^{\rho_p}  & \leq
\iint_{\mathcal{A}_M} |\alpha (\tau_{M})|^{\rho_p} + 2^{\rho_p-1}
\iint_{\mathcal{O}_M}|\alpha(M)|^{\rho_p}
 + 2^{\rho_p-1} C_1^{\rho_p}
\iint_{\mathcal{O}_M}| w_{M,{\mu}} - M|^{\rho_p}\\
&  \leq
 2^{\rho_p-1} \int_0^T \int_{\Omega} |\alpha(\tau_M)|^{\rho_p}+
C\| w_{M,{\mu}}\|_{L^{\rho_p}(0,T; L^{\rho_p}(\Omega))}^{\rho_p} +
 CM^{\rho_p} |\mathcal{O}_M| \\
 & \leq C_\mu + C + C \frac{M^{\rho_p}}{M^{p+2}},
 \end{aligned}$$ where the first inequality follows from the very definition of $\alpha_M$, the second one from trivial calculations, and the last one from estimates for $w_{M,{\mu}}$, for $|\mathcal{O}_M|$, and for $\alpha(\tau_M)$. Note that, since $\rho_p<2$, we have $M^{\rho_p}/ M^{p+2} \to 0 $ as $M \to +\infty$.

Altogether, we find $$\label{stima_n} \|\alpha_M (w_{M,{\mu}})\|_{L^{\rho_p}(0,T;
L^{\rho_p}(\Omega))} \leq C_\mu \quad \text{for all $M>0$,}$$ which yields, by comparison in , $$\label{stima_p} \|
\partial_t \chi_{M,{\mu}} \|_{L^{\rho_p} (0,T;W^{-2,\rho_p}(\Omega))}
 \leq C_\mu \quad \text{for all $M>0$,}$$ $W^{-2,\rho_p}(\Omega)$ denoting here the standard negative order Sobolev space.

Collecting estimates , , , and –, we then argue in the same way as in Section \[s:3.2\]. Thus, we conclude that there exist a subsequence $M_k \nearrow +\infty$ as $k \to +\infty$ (whose dependence on the index $\mu>0$ is not highlighted) and functions $(\chi_{\mu},w_{\mu})$ fulfilling – such that the functions $(\chi_{M_k,{\mu}}, w_{M_k,{\mu}})$ converge, as $k \to +\infty,$ to $(\chi_{\mu},w_{\mu})$ in the same sense as in - and , while, in place of , we only have $${\partial_t}\chi_{M_k,{\mu}}
  {\rightharpoonup}{\partial_t}\chi_{{\mu}} \quad
   \text{in \ $L^{\rho_p} (0,T;W^{-2,\rho_p}(\Omega))$,}$$ which is, anyway, sufficient for what follows. Furthermore, there exists some $\bar{\alpha}_\mu \in L^{\rho_p}(0,T;
L^{\rho_p}(\Omega))$ such that $$\alpha_M (w_{M_k,{\mu}}) {\rightharpoonup}\bar{\alpha}_\mu  \quad \text{in
$L^{\rho_p}(0,T; L^{\rho_p}(\Omega))$.}$$ Again, we prove that $$\label{post-2}
  \phi_{\mu}(\chi_{M_k,\mu}) \to \phi_{\mu}(\chi_{{\mu}}) \quad
   \text{in $L^2 (0,T;H)$}$$ and, proceeding as in Section \[s:3.2\], with we show that $\bar{\alpha}_\mu  =\alpha(w_\mu)$ and $$\alpha_{M_k}(w_{M_k,\mu}) \to \alpha(w_{\mu}) \qquad \text{in $L^1
(0,T; L^1 (\Omega))$.}$$ Having this, we conclude that the pair $(\chi_{\mu},w_{\mu})$ is solution to the PDE system $$\begin{aligned}
  & \label{1-var-better-mu-bis}
   \chi_t + A (\alpha(w)) =0 \qquad {\text{a.e.\ in}}\  \Omega \times (0,T)\,,\\
 &
 \label{2-var-better-mu-bis} A\chi + \phi_{{\mu}}(\chi)
  = w \qquad {\text{a.e.\ in}}\ \Omega \times (0,T)\,,\end{aligned}$$ supplemented with the initial condition .

. Finally, we take the limit $\mu\searrow 0$ in (the Cauchy problem for) –. Estimate , with $\alpha_M$ replaced by $\alpha$, holds for the sequence $\{(\chi_{\mu},w_{\mu})\}_{\mu}$ for a constant $C>0$ which is *independent* of the parameter $\mu>0$.

Furthermore, using the fact that system – has the same structure as –, we argue as in – and conclude that $$\exists\, C>0 \ \ \forall\, \mu>0\, : \quad \|\chi_{{\mu}}\|_{L^2
(0,T;W^{2,6}(\Omega))} +
 \| \phi_{\mu}(\chi_{\mu})\|_{L^2 (0,T; L^6(\Omega))}  \leq C.$$ From the bound for $(\alpha'(w_{\mu}))^{1/2} \nabla
w_{\mu}$ in $ L^2(0,T;H)$ (which follows from by applying once more Ioffe’s theorem), developing the very same calculations as throughout –, we find $$\label{giulio11}
\begin{aligned}
 \exists\, C>0 \ \ \forall\, \mu>0\,: \quad \|
\partial_t \chi_{\mu}\|_{L^{\eta_{p\sigma}}} (0,T;
{\mathcal{W}^{-2,{\kappa_p}}(\Omega)}) &  +
 \| \alpha(w_{\mu}) \|_{L^{\eta_{p\sigma}} (0,T; L^{\kappa_p}
 (\Omega))} \\ & +  \| \phi_{\mu}(\chi_{\mu}) \|_{L^{\sigma q_\sigma} (0,T;L^{6}(\Omega))} \leq
 C
 \end{aligned}$$ (where the indexes $\eta_{p\sigma}$ and $q_\sigma$ are as in and , respectively).

Thanks to the above estimates, we conclude that there exist a vanishing sequence ${\mu}_k \searrow 0$ and functions $(\chi,w)$ satisfying – such that $(\chi_{{\mu}_k}, w_{{\mu}_k})$ converges to $(\chi,w)$ in the topologies of –. We then pass to the limit as $k \to +\infty$ in  and, also in view of , infer that $\chi$ complies with the initial condition . Furthermore, we deduce from the strong convergence of $\chi_{\mu_k}$ to $\chi$ in $L^2
(0,T;H)$ that $\chi_{\mu_k} \to \chi$ almost everywhere in $\Omega
\times (0,T)$. Using the uniform convergence  of $\{\phi_{\mu_k}\}$ to $\phi$, we infer that $$\phi_{\mu_k} (\chi_{\mu_k}(x,t)) \to \phi(\chi(x,t)) \quad {{\text{for a.a.}}}\, (x,t)
 \in \Omega \times (0,T).$$ Then, taking into account the uniform integrability of $\{ \phi_{\mu}(\chi_{\mu_k})\}$ in $L^2 (0,T;H)$ (which follows from , noting that $\sigma q_\sigma>2$), in view of Theorem \[t:ds\] we obtain $$\label{step-fundamental}
  \phi_{\mu_k}(\chi_{\mu_k}) \to
\phi(\chi) \qquad \text{in $L^2 (0,T;H)$.}$$ Then, we again argue as in Section \[s:3.2\] and use to prove that $$\alpha(w_{\mu_k}) \to \alpha(w) \qquad \text{in $L^1 (0,T; L^1
(\Omega))$.}$$ Having this, we conclude that the pair $(\chi,w)$ is solution to Problem \[p:1\], which finishes the proof.

[99]{}

A.V. Babin and M.I. Vishik: *Attractors of evolution equations*. Translated and revised from the 1989 Russian original by Babin. Studies in Mathematics and its Applications, 25. North-Holland Publishing Co., Amsterdam, 1992.

J.M. Ball: Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations. *J. Nonlinear Sci.*, **7** (1997), 475–502.

V. Barbu: . Noordhoff, Leyden, 1976.

V. Barbu, P. Colli, G. Gilardi, and M. Grasselli: Existence, uniqueness, and long-time behaviour for a nonlinear Volterra integrodifferential equation. , **13** (2000), 1233–1262.

H. Brézis: . North Holland Math. Studies, 5. North-Holland, Amsterdam, 1973.

: . , **9** (1961), [795–801]{}.

: . , **2** (1958), [258–267]{}.

N. Dunford and J.T. Schwartz: . Interscience Publishers, New York, 1958.

: . .

M. E. Gurtin: Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. *Phys. D*, **92** (1996), [178–192]{}.

M. Hieber and J. Prüss: Heat kernels and maximal $L^p$-$L^q$ estimates for parabolic evolution equations. *Comm. Partial Differential Equations*, **22** (1997), 1647–1669.

A.D. Ioffe: On lower semicontinuity of integral functionals. I. *SIAM J. Control Optimization*, **15** (1977), 521–538.

J. Málek and D. Pražák: Large time behavior via the method of $l$-trajectories. *J. Differential Equations*, **181** (2002), 243–279.

A. Miranville and S. Zelik: Robust exponential attractors for Cahn-Hilliard type equations with singular potentials. *Math. Methods Appl. Sci.*, [**27**]{} (2004), 545–582.

: *Attractors for dissipative partial differential equations in bounded and unbounded domains*. .

: *The Cahn-Hilliard equation*. .

E. Rocca and G. Schimperna: Universal attractor for some singular phase transition systems. *Phys. D*, [**192**]{} (2004), 279–307.

R. Rossi: On two classes of generalized viscous Cahn-Hilliard equations. *Commun. Pure Appl. Anal.*, [**4**]{} (2005), 405–430.

R. Rossi: Global attractor for the weak solutions of a class of viscous Cahn-Hilliard equations. In: *Dissipative Phase Transitions*, pp. 247–268. Series on Advances in Mathematics for Applied Sciences, Vol. 71, World Sci. Publ., Hackensack, NJ, 2006.

R. Rossi, A. Segatti, and U. Stefanelli: Attractors for gradient flows of non convex functionals and applications to quasistationary phase field models. *Arch. Ration. Mech. Anal.*, [**187**]{} (2008), 91–135.

G. Schimperna: Global attractors for Cahn-Hilliard equations with nonconstant mobility. *Nonlinearity*, **20** (2007), 2365–2387.

A. Segatti: On the hyperbolic relaxation of the Cahn-Hilliard equation in 3D: approximation and long time behaviour. *Math. Models Methods Appl. Sci.*, **17** (2007), 411–437.

J. Simon: Compact sets in the space [$L^p(0,T;B)$]{}. *Ann. Mat. Pura Appl. (4)*, [**146**]{} (1987), 65–96.

: *Infinite-dimensional dynamical systems in mechanics and physics*, .

[^1]: *Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano. Via Bonardi, 9. I–20133 Milano, Italy. Email: [maurizio.grasselli@polimi.it]{}*

[^2]: *Laboratoire de Mathématiques et Applications–SP2MI, Université de Poitiers. Boulevard Marie et Pierre Curie–Téléport 2. F–86962 Chasseneuil Futuroscope Cedex, France. Email: [miranv@math.univ-poitiers.fr]{}*

[^3]: *Dipartimento di Matematica, Università di Brescia. Via Valotti 9. I–25133 Brescia, Italy.* E-mail: [riccarda.rossi@ing.unibs.it]{}

[^4]: *Dipartimento di Matematica “F.Casorati”, Università di Pavia. Via Ferrata, 1. I–27100 Pavia, Italy. Email: [giusch04@unipv.it]{}*

[^5]: All authors have been supported by the [project *Programma Galileo, Università Italo-Francese/Projet Galilée “Modelli matematici in scienza dei materiali/Modèles mathématiques en science des matériaux”.*]{} This paper was initiated during a stay of M.G., R.R., and G.S. in the *Laboratoire de Mathématiques et Applications* (Université de Poitiers), whose hospitality is gratefully acknowledged.
