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On a boundary value problem for wave equations: Existence uniqueness-asymptotic behavior. (English) Zbl 0859.35070
Summary: We consider the non-homogeneous boundary value problem: \[ u''(x,t)-\mu(t)\Delta u(x,t)=0\quad\text{in }\Omega\times]0,\infty[,\tag{\(*\)} \] \[ u=0\quad\text{on }\Gamma_0\times ]0,\infty[,\quad \mu(t){\partial u\over \partial\nu}+\delta(x)u'(x,t)=0\quad\text{on }\Gamma_1\times]0,\infty[, \] \[ u(x,0)=u^0(x),\;u'(x,0)=u^1(x)\quad\text{in }\Omega. \] With restrictions on \(\mu\) and \(\delta\) we prove existence and uniqueness of strong and weak solutions for problem \((*)\). We use the Galerkin method. We also prove exponential decay for the solutions of \((*)\).

MSC:
35L20 Initial-boundary value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
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