L∞ blow-up in the Jordan–Moore–Gibson–Thompson equation

The Jordan–Moore–Gibson–Thompson equation $\tau u_{ttt}+\alpha u_{tt}=\beta \Delta u_t+\gamma \Delta u+{\left(f\left(u\right)\right)}_{tt}$is considered in a smoothly bounded domain $\Omega \subset R^n$ with $n\le 3$, where $\tau >0,\beta >0,\gamma >0$, and $\alpha \in R$.

Firstly, it is seen that under the assumption that $f\in C^3\left(R\right)$ is such that $f\left(0\right)=0$, 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 $\text{if}{T}_{max}<\infty ,\text{then}\underset{t\nearrow T_{max}}{lim\; sup}{\Vert u(\cdot ,t)\Vert }_{L^{\infty }\left(\Omega \right)}=\infty .(\star )$This is used to, secondly, make sure that if additionally $f$ is convex and grows superlinearly in the sense that $f^{\prime \prime }\ge 0\text{on}R\text{,}\frac{f\left(\xi \right)}{\xi }\to +\infty \text{as}\xi \to +\infty \text{and}{\int }_{{\xi }_0}^{\infty }\frac{d\xi }{f\left(\xi \right)}<\infty \text{for some}{\xi }_0>0\text{,}$then for some initial data the above solution must undergo some finite-time $L^{\infty }$ blow-up in the style described in ($\star$).