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Convergence to equilibrium for the damped semilinear wave equation with critical exponent and dissipative boundary condition. (English) Zbl 1120.35025
The authors consider the damped semilinear wave equation \(u_{tt} + u_{t} - \Delta u + f(x,u) = 0\) on a bounded domain \(\Omega \subset \mathbb R^{3}\) subject to the dissipative boundary condition \(\partial_{\nu}u + u+ u_{t} = 0\) (\(t > 0\), \(x\in \partial \Omega\)). The nonlinearity is assumed to be analytic in \(u\) and satisfy growth and dissipativity conditions. Using a suitable modification of Simon-Lojasiewicz inequality, the authors prove that each solution of the above problem converge to an equilibrium as time goes to infinity. The novelty of this theorem, compared to previous convergence results for the damped wave equation, is that dissipative boundary condition is treated and the critical growth of \(f\) is allowed.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35L70 Second-order nonlinear hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
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