Almost global existence for nonlinear wave equations in an exterior domain in two space dimensions

TL;DR

This paper proves that small, smooth initial data for nonlinear wave equations in an exterior domain in two dimensions lead to solutions that exist for a long time, similar to the Cauchy problem, despite weak decay properties.

Contribution

It extends lifespan estimates for nonlinear wave equations to exterior domains in two dimensions, showing almost global existence under small initial data.

Findings

Lifespan of solutions matches that of the Cauchy problem.

Solutions exhibit weak decay but still have long existence times.

Results apply to nonlinear wave systems in exterior domains.

Abstract

In this paper we deal with the exterior problem for a system of nonlinear wave equations in two space dimensions, assuming that the initial data is small and smooth. We establish the same type of lower bound of the lifespan for the problem as that for the Cauchy problem, despite of the weak decay property of the solution in two space dimensions.