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Remarks on weak stabilization of semilinear wave equations. (English) Zbl 0988.35029
Summary: If a second-order semilinear conservative equation with esssentially oscillatory solutions such as the wave equation is perturbed by a possibly nonmonotone damping term, which is effective in a nonnegligible subregion for at least one sign of the velocity, all solutions of the perturbed system converge weakly to zero as time tends to infinity. We present here a simple and natural method of proof of this kind of property, implying as a consequence some recent very general results of Judith Vancostenoble.
MSC:
35B40Asymptotic behavior of solutions of PDE
35L90Abstract hyperbolic equations
35B35Stability of solutions of PDE
35L55Higher order hyperbolic systems
35L70Nonlinear second-order hyperbolic equations