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Wave equation in domains with non locally reacting boundary. (English) Zbl 1249.35221
The authors seek a pair of the functions \(u,\delta \) satisfying \(u_{tt}-M(\| u\| ^2_{L^2_{\Omega }})\Delta u+\alpha u_t+\beta | u_t| ^{\rho }u_t=0\) in \(\Omega \times (0,\infty )\), \(u=0\) on \(\Gamma _0\times (0,\infty )\), \(\frac {\partial u}{\partial n}=\delta _t\) on \(\Gamma _1\times (0,\infty )\), \(f\delta _{tt}-k^2\Delta _{\Gamma }\delta +g\delta _t+h\delta =-u_t\) on \(\Gamma _1\times (0,\infty )\), \(\delta =0\) on \(\partial \Gamma _1\times (0,\infty )\), \(u(x,0)=u_0(x)\), \(u_t(x,0)=u_1(x)\), \(x\in \Omega \), \(\delta (x,0)=\delta _0(x)\) and \(\delta _t(x,0)=\frac {\partial u_0}{\partial n}(x)\), \(x\in \Gamma _1\), where \(\Delta _{\Gamma }\) is the Laplace-Beltrami operator, \(\Omega \) is a bounded region in \(\mathbb {R}^n\) and \(\Gamma _0\cup \Gamma _1=\partial \Omega \). Using approximations in a finite dimensional spaces the authors prove existence and uniqueness of a generalized solution together with asymptotic stability of the energy to the problem. Moreover, they give existence of at least one solution to the similar problem, when \(M\equiv 1\), \(\alpha =\beta =0\), the second boundary condition on \(u\) is replaced by \(\frac {\partial u}{\partial n}=\phi (\| \delta \| ^2_{L^2(\Gamma _1)})\delta _t\) and the second initial data for \(\delta \) is replaced by \(\delta _t(x,0)=\frac {1}{\phi (\| \delta _0 \| ^2_{L^2(\Gamma _1)})}\frac {\partial u_0}{\partial n}(x)\).

35L71 Second-order semilinear hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
35Q35 PDEs in connection with fluid mechanics
35R01 PDEs on manifolds