$L^\infty$ blow-up in the Jordan-Moore-Gibson-Thompson equation

Abstract

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.