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Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity. (English) Zbl 0939.35122
The paper deals with the initial boundary value problem for the equation \( u_{tt}+cu_{t}= \triangle u+f(x,u)\) in \(\mathbb{R}\times \Omega ,\) where \(\Omega \) is a bounded smooth domain in \(\mathbb{R}^{N}\), \(c\) is a positive constant, and \(f\) is an analytic (in \(u\)) function satisfying some growth conditions. The authors prove, that if the trajectory of a solution \(u\) is bounded in \(H^{0}_{1}(\Omega) \times L^{2}(\Omega),\) then \(u\) converges to a solution of the appropriate stationary problem. Some examples and some more general variants of the result are also given.

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