Criteria of the existence of global solutions to semilinear wave equations with first-order derivatives on exterior domains

TL;DR

This paper establishes criteria for the existence of global solutions to semilinear wave equations with first-order derivatives on exterior domains, using energy estimates and Sobolev embeddings, and provides lifespan estimates for specific cases.

Contribution

It provides new criteria for global existence of solutions to semilinear wave equations on exterior domains, including radial cases and obstacle considerations, using advanced analytical techniques.

Findings

Global existence criteria depend on the dimension and nonlinearity order.

Criteria are verified for radial solutions with obstacle assumptions.

Sharp lifespan estimates are obtained for specific nonlinearities and initial data.

Abstract

We study the existence of global solutions to semilinear wave equations on exterior domains $\mathbb{R}^n\setminus\mathcal{K}$, $n\geq2$, with small initial data and nonlinear terms $F(\partial u)$ where $F\in C^\kappa$ and $\partial^{\leq\kappa}F(0)=0$. If $n\geq2$ and $\kappa>n/2$, criteria of the existence of a global solution for general initial data are provided, except for non-empty obstacles $\mathcal{K}$ when $n=2$. For $n\geq3$ and $1\leq\kappa\leq n/2$, we verify the criteria for radial solutions provided obstacles $\mathcal{K}$ are closed balls centered at origin. These criteria are established by local energy estimates and the weighted Sobolev embedding including trace estimates. Meanwhile, for the sample choice of the nonlinear term and initial data, sharp estimates of lifespan are obtained.