Lower bound of the lifespan of solutions to semilinear wave equations in an exterior domain

TL;DR

This paper establishes a sharp lower bound on the lifespan of classical solutions to semilinear wave equations in a three-dimensional exterior domain with small initial data, extending previous results to exterior domains.

Contribution

It provides a detailed lower bound estimate for solution lifespan in exterior domains, generalizing classical results from the Cauchy problem to more complex geometries.

Findings

Lower bound estimate for solution lifespan in exterior domains

The estimate is sharp in certain cases

Extension of classical lifespan results to exterior obstacle problems

Abstract

We consider the Cauchy-Dirichlet problem for semilinear wave equations in a three space dimensional domain exterior to a bounded and non-trapping obstacle. We obtain a detailed estimate for the lower bound of the lifespan of classical solutions when the size of initial data tends to zero, in a similar spirit to that of the works of John and H\"ormander where the Cauchy problem was treated. We show that our estimate is sharp at least for some special case.