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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
##### Keywords:
Galerkin method; exponential decay