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Stabilization of trajectories for some weakly damped hyperbolic equations. (English) Zbl 0535.35006
We establish the extinction of self-oscillations for two classes of weakly damped hyperbolic equations in a bounded domain. For equations of the first class, which are autonomous, we prove the convergence to an equilibrium even though the damping term vanishes identically in a set of positive measure inside the domain. The second class of equations consists in quasi-autonomous periodic equations with a local, nonlinear damping term which is strictly increasing for small values of \(u_ t:\) we establish that any strong periodic solution is unique and globally asymptotically stable, while uniqueness can fail for weak periodic solutions.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35L70 Second-order nonlinear hyperbolic equations
35B10 Periodic solutions to PDEs
35B35 Stability in context of PDEs
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