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

The Jordan-Moore-Gibson-Thompson equation $$\tau u_{ttt}+\alpha u_{tt}=\beta \mathrm{\Delta }{u}_t+\gamma \mathrm{\Delta }u+(f(u){)}_{tt}$$ is considered in a smoothly bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ 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 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 on $\mathbb{R$,} \qquad \frac{f(ξ)}ξ \to +\infty as $ξ\to +\infty$ \qquad and \qquad \int_{ξ_0}^\infty \frac{dξ}{f(ξ)} < \infty for some $ξ_0>0$, then for some initial data the above solution must undergo some finite-time $L^{\mathrm{\infty }}$ blow-up in the style described above.