JOURNAL OF DIFFERENTIAL EQUATIONS | 卷:252 |
Asymptotics for some semilinear hyperbolic equations with non-autonomous damping | |
Article | |
Cabot, A.1  Frankel, P.1  | |
[1] Univ Montpellier 2, Dept Math, Montpellier, France | |
关键词: Semilinear evolution problem; Dissipative hyperbolic equation; Non-autonomous damping; Asymptotic behavior; Rate of convergence; | |
DOI : 10.1016/j.jde.2011.09.012 | |
来源: Elsevier | |
【 摘 要 】
Let V and H be Hilbert spaces such that V subset of H subset of V' with dense and continuous injections. Consider a linear continuous operator A: V V' which is assumed to be symmetric, monotone and semi-coercive. Given a function f: V -> H and a map gamma is an element of Wloc 1.1 (R(+), R(+)) such that lim(t) -> + infinity gamma(t) = 0, our purpose is to study the asymptotic behavior of the following semilinear hyperbolic equation d(2)u/dt(2) (T) + gamma(t) du/dt (t) + Au(t) + f(u(t)) = 0, t >= 0. (E) The nonlinearity f is assumed to be monotone and conservative. Condition integral(-) (+infinity) gamma(t)dt = + infinity guarantees that some suitable energy function tends toward its minimum. The main contribution of this paper is to provide a general result of convergence for the trajectories of (E): if the quantity gamma(t) behaves as k/t(alpha), for some alpha is an element of [0, 1], k > 0 and t large enough, then u(t) weakly converges in V toward an equilibrium as t -> + infinity. Strong convergence in V holds true under compactness or symmetry conditions. We also give estimates for the speed of convergence of the energy under some ellipticity-like conditions. The abstract results are applied to particular semilinear evolution problems at the end of the paper. (C) 2011 Elsevier Inc. All rights reserved.
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