Asymptotics for some semilinear hyperbolic equations with non-autonomous damping

Let V and H   be Hilbert spaces such that V⊂H⊂V′V⊂H⊂V′ with dense and continuous injections. Consider a linear continuous operator A:V→V′A:V→V′ which is assumed to be symmetric, monotone and semi-coercive. Given a function f:V→Hf:V→H and a map γ∈Wloc1,1(R+,R+) such that limt→+∞γ(t)=0, our purpose is to study the asymptotic behavior of the following semilinear hyperbolic equationd2udt2(t)+γ(t)dudt(t)+Au(t)+f(u(t))=0,t⩾0. The nonlinearity f   is assumed to be monotone and conservative. Condition ∫0+∞γ(t)dt=+∞ 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 γ(t)γ(t) behaves as k/tαk/tα, for some α∈]0,1[α∈]0,1[, k>0k>0 and t   large enough, then u(t)u(t) weakly converges in V   toward an equilibrium as t→+∞t→+∞. 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.