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Blow-up for the wave equation with hyperbolic dynamical boundary conditions, interior and boundary nonlinear damping and sources. (English) Zbl 1480.35065

Summary: 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 \(\mathbb{R}^N,\) \(N\geq 2,\) \(\Gamma = \partial\Omega,\) \((\Gamma_0,\Gamma_1)\) is a partition of \(\Gamma,\) \(\Gamma_1 \neq \emptyset\) 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.

MSC:

35B44 Blow-up in context of PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L71 Second-order semilinear hyperbolic equations
35Q74 PDEs in connection with mechanics of deformable solids
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