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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.

##### MSC:
 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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