The energy decay and asymptotics for a class of semilinear wave equations in two space dimensions

TL;DR

This paper studies the long-term behavior of small solutions to certain semilinear wave equations in two dimensions, demonstrating global existence, energy decay, and asymptotic descriptions for cubic nonlinearities.

Contribution

It establishes the global existence and asymptotic behavior of solutions for a class of semilinear wave equations with cubic nonlinearity in two space dimensions.

Findings

Solutions exist globally for small initial data.

Energy decays over time in dissipative cases.

Asymptotic descriptions of solutions as time approaches infinity.

Abstract

We consider semilinear wave equations with small initial data in two space dimensions. For a class of wave equations with cubic nonlinearity, we show the global existence of small amplitude solutions, and give an asymptotic description of the solution as $t \to \infty$ uniformly in $x \in {\mathbb R}^2$. In particular, our result implies the decay of the energy when the nonlinearity is dissipative.