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

Abstract

The Jordan-Moore-Gibson-Thompson equation \[ τu_{ttt} + αu_{tt} = βΔu_t + γΔu + (f(u))_{tt} \] is considered in a smoothly bounded domain $\Omega \subset {\mathbb{R}}^n$ with $n\le 3$, 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 \[ \mbox{if $T_{max}<\infty$, then } \limsup_{t\nearrow T_{max}} \|u(\cdot,t)\|_{L^\infty(Ω)}=\infty. \] This is used to, secondly, make sure that if additionally $f$ is convex and grows superlinearly in the sense that \[ f''\ge 0 \mbox{ on $\mathbb{R}$,} \qquad \frac{f(ξ)}ξ \to +\infty \mbox{ as $\xi \to +\infty$} \qquad \mbox{and} \qquad \int_{ξ_0}^\infty \frac{dξ}{f(ξ)} < \infty \mbox{ for some ${\xi }_0>0$,} \] then for some initial data the above solution must undergo some finite-time $L^{\infty }$ blow-up in the style described above.