Lifespan of Classical Solutions to Quasilinear Wave Equations Outside of a Star-Shaped Obstacle in Four Space Dimensions

TL;DR

This paper establishes a lower bound for the lifespan of classical solutions to quasilinear wave equations outside a star-shaped obstacle in four dimensions, matching the sharp bound known for the Cauchy problem.

Contribution

It introduces new $L^{ abla}_{t}L^{2}_{x}$ and weighted $L^{2}_{t,x}$ estimates for the obstacle problem, extending lifespan results to this setting.

Findings

Lifespan lower bound $T_{ ext{eps}} extgreater ext{exp}(c/ ext{eps}^2)$

New estimates for quasilinear wave equations outside obstacles

Sharp lifespan estimate matching the Cauchy problem

Abstract

We study the initial-boundary value problem of quasilinear wave equations outside of a star-shaped obstacle in four space dimensions, in which the nonlinear term under consideration may explicitly depend on the unknown function itself. By some new $L^{\infty}_{t}L^{2}_{x}$ and weighted $L^{2}_{t,x}$ estimates for the unknown function itself, together with energy estimates and KSS estimates, for the quasilinear obstacle problem we obtain a lower bound of the lifespan $T_{\varepsilon}\geq \exp{(\frac{c}{\varepsilon^2})}$, which coincides with the sharp lower bound of lifespan estimate for the corresponding Cauchy problem.