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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 $$\delta _t(x,0)=\frac {1}{\phi (\| \delta _0 \| ^2_{L^2(\Gamma _1)})}\frac {\partial u_0}{\partial n}(x)$$.

##### MSC:
 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