Asymptotic for a semilinear hyperbolic equation with asymptotically vanishing damping term, convex potential, and integrable source

TL;DR

This paper studies the long-term behavior of solutions to a semilinear hyperbolic equation with a damping term that vanishes asymptotically, establishing conditions for solutions to converge to stationary solutions.

Contribution

It provides new sufficient conditions on the source term g(t) that guarantee convergence of solutions to stationary solutions in a hyperbolic PDE with asymptotically vanishing damping.

Findings

Solutions converge weakly or strongly under certain conditions.

Convergence depends on the decay rate of the damping term.

Results apply to equations with convex potential and integrable source.

Abstract

We investigate the long time behavior of solutions to semilinear hyperbolic equation (E$_{\alpha}$): $ u^{\prime\prime}(t)+\gamma(t)u^{\prime}(t)+Au(t)+f(u(t))=g(t),~t\geq0, $ where $A$ is a self-adjoint nonnegative operator, $f$ a function which derives from a convex function, and $\gamma$ a nonnegative function which behaviors, for $t$ large enough, as $\frac{K}{t^{\alpha}}$ with $K>0$ and $\alpha \in\lbrack0,1[.$ We obtain sufficient conditions on the source term $g(t),$ ensuring the weak or the strong convergence of any solution $u(t)$ of (E$_{\alpha}$) as $t\rightarrow+\infty$ to a solution of the stationary equation $Av+f(v)=0$ if one exists.