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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

##### MSC:
 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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