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On the quenching of solutions of the wave equation with a nonlinear boundary condition. (English) Zbl 0698.35088
We consider the initial-boundary value problem (P): \(u_{tt}=\Delta u\) in \(\Omega\) \(\times [0,T)\), \(u(x,0)=u_ 0(x)\), \(u_ t(x,0)=\dot u_ 0(x)\), \(u(x,t)=0\) on \(\Gamma_ 0\times [0,T)\), \(\partial u(x,t)/\partial n=\phi (u(x,t))\) on \(\Gamma_ 1\times [0,T)\); where \(\partial \Omega =\Gamma_ 0\cup \Gamma_ 1\), \(\Omega\) is a “smooth” bounded domain in \({\mathbb{R}}^ n\), \(n=2\) or 3, and \(\phi\) : (-\(\infty,1)\to (0,\infty)\) with \(\phi\) is monotone increasing, \(\phi \in C^ 3(-\infty,1)\), and \(\phi (1^-)=+\infty\). For appropriate initial data, it is shown that problem (P) admits local strong solutions. Such solutions are continuous on \(\Gamma_ 1\). Moreover, under certain geometric condition on \(\Omega\) it is shown that there exists at least one point \(x_ 0\in \Gamma_ 1\) such that \(u(x_ 0,t)\) reaches one in finite-time or infinite-time depending on the integrability of \(\phi\) near \(u=1\).
Reviewer: M.A.Rammaha

35L05 Wave equation
35L20 Initial-boundary value problems for second-order hyperbolic equations
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
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