  This paper deals with the asymptotic behavior as $t\rightarrow T<\infty$ of
all weak (energy) solutions of a class of equations with the following model
representative: \begin{equation*}
  (|u|^{p-1}u)_t-\Delta_p(u)+b(t,x)|u|^{\lambda-1}u=0 \quad
(t,x)\in(0,T)\times\Omega,\,\Omega\in{R}^n,\,n>1, \end{equation*} with
prescribed global energy function \begin{equation*}
E(t):=\int_{\Omega}|u(t,x)|^{p+1}dx+
\int_0^t\int_{\Omega}|\nabla_xu(\tau,x)|^{p+1}dxd\tau \rightarrow\infty\ \text{
as }t\rightarrow T. \end{equation*} Here
$\Delta_p(u)=\sum_{i=1}^n\left(|\nabla_xu|^{p-1}u_{x_i}\right)_{x_i}$, $p>0$,
$\lambda>p$, $\Omega$ is a bounded smooth domain, $b(t,x)\geq0$. Particularly,
in the case \begin{equation*} E(t)\leq
F_\mu(t)=\exp\left(\omega(T-t)^{-\frac1{p+\mu}}\right)\quad\forall\,t<T,\,\mu>0,\,\omega>0,
\end{equation*} it is proved that solution $u$ remains uniformly bounded as
$t\rightarrow T$ in an arbitrary subdomain
$\Omega_0\subset\Omega:\overline{\Omega}_0\subset\Omega$ and the sharp upper
estimate of $u(t,x)$ when $t\rightarrow T$ has been obtained depending on
$\mu>0$ and $s=dist(x,\partial\Omega)$. In the case $b(t,x)>0$
$\forall\,(t,x)\in(0,T)\times\Omega$ sharp sufficient conditions on
degeneration of $b(t,x)$ near $t=T$ that guarantee mentioned above boundedness
for arbitrary (even large) solution have been found and the sharp upper
estimate of a final profile of solution when $t\rightarrow T$ has been
obtained.
