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Well-posedness and asymptotic behavior of a nonautonomous, semilinear hyperbolic-parabolic equation with dynamical boundary condition of memory type. (English) Zbl 1286.35042

Summary: We consider a nonautonomous, semilinear, hyperbolic-parabolic equation subject to a dynamical boundary condition of memory type. First we prove the existence and uniqueness of global bounded solutions having relatively compact range in the natural energy space. Under the assumption that the nonlinear term \(f\) is real analytic, we then derive an appropriate Lyapunov energy and we use the Łojasiewicz-Simon inequality to show the convergence of global weak solutions to single steady states as time tends to infinity. Finally, we provide an estimate for the convergence rate.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35M12 Boundary value problems for PDEs of mixed type
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