  The Jordan-Moore-Gibson-Thompson equation
  \[
  \tau u_{ttt} + \alpha u_{tt} = \beta \Delta u_t + \gamma \Delta u +
(f(u))_{tt} \] is considered in a smoothly bounded domain $\Omega
\subset\mathbb{R}^n$ with $n\leq 3$, where $\tau>0,\beta>0,\gamma>0$, and
$\alpha\in\mathbb{R}$.
  Firstly, it is seen that under the assumption that $f\in C^3(\mathbb{R})$ is
such that $f(0)=0$, gradient blow-up phenomena cannot occur in the sense that
for any appropriately regular initial data, within a suitable framework of
strong solvability, an associated Dirichlet type initial-boundary value problem
admits a unique solution $u$ on a maximal time interval $(0,T_{max})$ which is
such that \[
  \mbox{if $T_{max}<\infty$, then } \limsup_{t\nearrow T_{max}}
\|u(\cdot,t)\|_{L^\infty(\Omega)}=\infty. \] This is used to, secondly, make
sure that if additionally $f$ is convex and grows superlinearly in the sense
that \[
  f''\ge 0
  \mbox{ on $\mathbb{R}$,}
  \qquad
  \frac{f(\xi)}{\xi} \to +\infty
  \mbox{ as $\xi\to +\infty$}
  \qquad \mbox{and} \qquad
  \int_{\xi_0}^\infty \frac{d\xi}{f(\xi)} < \infty
  \mbox{ for some $\xi_0>0$,} \] then for some initial data the above solution
must undergo some finite-time $L^\infty$ blow-up in the style described above.
