Long time behavior for a semilinear hyperbolic equation with asymtotically vanishing damping term and convex potential

TL;DR

This paper analyzes the long-term behavior of solutions to a semilinear hyperbolic equation with a damping term that diminishes over time, proving convergence to equilibrium under specific damping conditions.

Contribution

It establishes the weak convergence of solutions to equilibrium for a class of hyperbolic equations with asymptotically vanishing damping, answering an open question from prior research.

Findings

Solutions converge weakly to equilibrium points.

Convergence holds when damping behaves like K/t^α with 0<α<1.

Results extend understanding of long-term dynamics in damped hyperbolic equations.

Abstract

We investigate the asymptotic behavior, as t goes to infinity, for a semilinear hyperbolic equation with asymptotically smal dissipation and convex potential. We prove that if the damping term behaves like K/t^\alpha for t large enough, k>0 and 0</alpha<1 then every global solution converges weakly to an equilibrium point. This result is a positive answer to a question left open in the paper [A. Cabot and P. Frankel, Asymptotics for some semilinear hyperbolic equation with non-autonomous damping. J. Differential Equations 252 (2012) 294-322.]