Asymptotics for some semilinear hyperbolic equations with non-autonomous damping
Résumé
Let $V$ and $H$ be Hilbert spaces such that $V\subset H\subset V'$ with dense and continuous injections. Consider a linear continuous operator $A:V\to V'$ which is assumed to be symmetric, monotone and semi-coercive. Given a function $f:V\to H$ and a map $\gamma\in W^{1,1}_{loc}(\mathbb{R}_+,\mathbb{R}_+)$ such that $\lim_{t\to+\infty}\gamma (t)=0$, our purpose is to study the asymptotic behavior of the following semilinear hyperbolic equation $$(E) \qquad \frac{d^2u}{dt^2}(t)+\gamma(t)\frac{du}{dt}(t)+Au(t)+f(u(t))=0,\quad t\geq 0.$$ The nonlinearity $f$ is assumed to be monotone and conservative. Condition $\int_0^{+\infty} \gamma (t)\, dt =~+\infty$ 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 $\alpha\in ]0,1[$, $k>0$ and $t$ large enough, then $u(t)$ weakly converges in $V$ toward an equilibrium as $t\to +\infty$. 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.
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