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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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[1] Amerio, L.; Prouse, G., Uniqueness and almost-periodicity theorems for a nonlinear wave equation, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 46, 1-8 (1969) · Zbl 0176.40403
[2] Brézis, H., Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert (1973), North-Holland: North-Holland Amsterdam/London · Zbl 0252.47055
[3] Dafermos, C. M.; Slemrod, M., Asymptotic behavior of nonlinear contraction semi-groups, J. Funct. Anal., 13, 97-106 (1973) · Zbl 0267.34062
[4] Dafermos, C. M., Contraction semi-groups and trend to equilibrium in continuum mechanics, (Lecture Notes in Mathematics, Vol. 503 (1976), Springer-Verlag: Springer-Verlag New York/Berlin), 295-306 · Zbl 0345.47032
[5] Dafermos, C. M., Asymptotic behavior of solutions of evolution equations, (Crandall, M. G., Nonlinear Evolution Equations (1978), Academic Press: Academic Press New York), 103-123 · Zbl 0499.35015
[6] Haraux, A., Comportement à l’infini pour une équation des ondes non linéaire dissipative, C. R. Acad. Sci. Paris Ser. A, 287, 507-509 (1978) · Zbl 0396.35065
[7] Haraux, A., Almost-periodic forcing for a wave equaiton with a nonlinear, local damping term, (Proc. Roy. Soc. Edinburgh Sect. A, 94 (1983)), 195-212 · Zbl 0589.35076
[8] Haraux, A., Damping out of transient states for some semi-linear, quasi-autonomous systems of hyperbolic type, Rend. Acad. Naz. Sci. XL, VII 7, 89-136 (1983) · Zbl 0556.35093
[9] Haraux, A., Nonlinear Evolution Equations: Global Behavior of Solutions, (Lecture Notes in Mathematics, Vol. 841 (1981), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0583.35007
[10] Haraux, A., On a uniqueness theorem of L. Amerio and G. Prouse, (Proc. Roy. Soc. Edinburgh, 96A (1984)), 221-230 · Zbl 0555.35090
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