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# Blow–up for the wave equation with hyperbolic dynamical boundary conditions, interior and boundary nonlinear damping and sources

The work was realized within the auspices of the INdAM – GNAMPA Projects Equazioni alle derivate parziali: Problemi e Modelli (Prot_U-UFMBAZ-2020-000761), and it was also supported by Progetto Equazione delle onde con condizioni acustiche, finanziato con il Fondo Ricerca di Base, 2019, della Università degli Studi di Perugia and by Progetti Equazioni delle onde con condizioni iperboliche ed acustiche al bordo, finanziati con i Fondi Ricerca di Base 2017 and 2018, della Università degli Studi di Perugia
• The aim of this paper is to give global nonexistence and blow–up results for the problem

$\begin{cases} u_{tt}-\Delta u+P(x,u_t) = f(x,u) \qquad &\text{in$(0,\infty)\times\Omega$,}\\ u = 0 &\text{on$(0,\infty)\times \Gamma_0$,}\\ u_{tt}+\partial_\nu u-\Delta_\Gamma u+Q(x,u_t) = g(x,u)\qquad &\text{on$(0,\infty)\times \Gamma_1$,}\\ u(0,x) = u_0(x),\quad u_t(0,x) = u_1(x) & \text{in$\overline{\Omega}$,} \end{cases}$

where $\Omega$ is a bounded open $C^1$ subset of ${\mathbb R}^N$, $N\ge 2$, $\Gamma = \partial\Omega$, $(\Gamma_0,\Gamma_1)$ is a partition of $\Gamma$, $\Gamma_1\not = \emptyset$ being relatively open in $\Gamma$, $\Delta_\Gamma$ denotes the Laplace–Beltrami operator on $\Gamma$, $\nu$ is the outward normal to $\Omega$, and the terms $P$ and $Q$ represent nonlinear damping terms, while $f$ and $g$ are nonlinear source terms. These results complement the analysis of the problem given by the author in two recent papers, dealing with local and global existence, uniqueness and well–posedness.

Mathematics Subject Classification: Primary: 35L05, 35L10, 35L20; Secondary: 35D30, 35Q74.

 Citation: • • Figure 1.  The regions covered by (6), and their subregions in which (14) holds or not, in dimensions $N = 2,3,4$

Table 1.  (18) vs. (19) for the model classes A–D)

 Case (16) A) $\gamma = 0$, $\delta = 0$ B) $\gamma = 0$, $\delta>0$ C) $\gamma>0$, $\delta = 0$ D) $\gamma>0$, $\delta>0$ (18) $\checkmark$ $q = 2$ $p = 2$ $p = 2$, $q = 2$ (19) ${\mathcal {X}}$ $q>2$ $p>2$ $p>2$, $q>2$

Table 2.  (14) vs. (22) for the model problems Ca–Cd)

 $\gamma>0 = \delta$ $\alpha = \beta = 0$ $\alpha = 0<\beta$ $\alpha>0 = \beta$ $\alpha,\beta>0$ (14) $p = 2$ $p = 2$ $p\le\overline{m}$ $p\le\overline{m}$ (22) $p>2$ $p>2$, $\overline{\mu}<1+p/2$ $p>\overline{m}$ $p>\overline{m}$, $\overline{\mu}<1+p/2$

Table 3.  (14) vs (26) for the model problems Da–Dd)

 $\gamma,\delta>0$ $\alpha = \beta = 0$ $\alpha = 0<\beta$ $\alpha>0 = \beta$ $\alpha,\beta>0$ (14) $p = 2$,$q = 2$ $p = 2$, $q\le \overline{\mu}$ $p\le\overline{m}$, $q = 2$ $p\le\overline{m}$, $q\le\overline{\mu}$ (26) $p>2$, $q>2$ $p>2$, $q> \begin{cases}2 & \text { if } \bar{\mu} <1+\frac{p}{2} \\ \bar{\mu} & \text { if } \bar{\mu} \geq 1+\frac{p}{2}\end{cases}$ $p>\overline{m}$, q>2 $p>\overline{m}$, $q> \begin{cases}2 & \text { if } \bar{\mu} <1+\frac{p}{2} \\ \bar{\mu} & \text { if } \bar{\mu} \geq 1+\frac{p}{2}\end{cases}$
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