Lifespan estimates for $2$-dimensional semilinear wave equations in asymptotically Euclidean exterior domains

TL;DR

This paper investigates lifespan estimates for small-data semilinear wave equations in two-dimensional asymptotically Euclidean exterior domains, showing that the lifespan bounds are similar to the Cauchy problem with minimal loss, even with obstacles and flat metrics.

Contribution

It extends lifespan estimates to 2D exterior domains with obstacles and flat metrics, handling general obstacles and reducing the problem to compact perturbations.

Findings

Lifespan bounds match those of the Cauchy problem for 1<p≤pc(2).

Logarithmic harmonic function growth does not affect lifespan estimates for 2<p≤pc(2).

The approach applies to flat metrics and general obstacles, including bounded and simply connected.

Abstract

In this paper we study the initial boundary value problem for two-dimensional semilinear wave equations with small data, in asymptotically Euclidean exterior domains. We prove that if $1<p\le p_c(2)$, the problem admits almost the same upper bound of the lifespan as that of the corresponding Cauchy problem, only with a small loss for $1<p\le 2$. It is interesting to see that the logarithmic increase of the harmonic function in $2$-D has no influence to the estimate of the upper bound of the lifespan for $2<p\le p_c(2)$. One of the novelties is that we can deal with the problem with flat metric and general obstacles (bounded and simple connected), and it will be reduced to the corresponding problem with compact perturbation of the flat metric outside a ball.