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

arXiv:2610.33213v1 Announce Type: new Abstract: We investigate the fourth-order Jordan-Moore-Gibson-Thompson equation \bee u_{tttt}+\al u_{ttt}+\bet u_{tt}-\gamma \Del u_{tt}-\del \Del u_{t}-\varrho \Del u=(f(u))_{ttt}+(f(u))_{tt} \eee under homogeneous Dirichlet boundary conditions in a smooth bounded domain $\Om \subset \R^n$ with $n\leq 3$, where $\al \in \R$ and $\bet,\gamma,\del,\varrho>0$. First, for any suitably regular initial data and assuming that $f\in C^4(\R)$ with $f(0)=0$, we prove that the corresponding initial-boundary value problem admits a unique maximal strong solution $u$ on $(0, \tm)$. Second, by combining three time-integrated formulations with a series of resolvent-regularized energy estimates, we show that gradient blow-up is ruled out in the sense that \be{EE} \tm<\infty \quad\,\, \Rightarrow\quad\,\, \limsup_{t\nearrow \tm}\|u(\cd,t)\|_{L^{\infty}(\Om)}=\infty. \tag{$\star$} \ee Finally, assuming in addition that $f$ is convex and grows superlinearly in the sense that \bee f''(\xi)\geq 0 \;\; \mbox{ for all}\;\;\xi \in\R, \qquad \frac{f(\xi)}{\xi}\rightarrow +\infty\;\;\mbox{ as}\;\; \xi\rightarrow +\infty , \qquad \int_{\xi_{0}}^{\infty}\frac{d\xi}{f(\xi)}<\infty \;\; \mbox{ for some}\;\; \xi_{0}>0, \eee we employ the extensibility criterion (\ref{EE}) to prove that, for a suitable class of initial data, the solution blows up in finite time in the $L^\infty$-norm. To the best of our knowledge, this provides the first rigorous description of finite-time blow-up for a genuinely nonlinear equation of fourth order in time.

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