Long-Time Existence of Quasilinear Wave Equations Exterior to Star-shaped Obstacle in $2\mathbf{D}$

TL;DR

This paper proves long-time existence for small data solutions of quasilinear wave equations outside star-shaped obstacles in 2D, establishing decay estimates and lifespan bounds that are nearly optimal.

Contribution

It introduces a Morawetz type energy estimate and new weighted $L^2$ estimates for exterior wave equations, advancing understanding of lifespan and decay in 2D exterior domains.

Findings

Established $t^{-1/2}$ decay estimates inside the cone.

Derived lifespan relation $oxed{ ext{T}_ ext{ε} ext{ satisfying } ext{ε}^2 T_ ext{ε} ext{ln}^3 T_ ext{ε} = A}$.

Proved near-sharp lifespan bounds with logarithmic loss.

Abstract

In this paper, we study the long-time existence result for small data solutions of quasilinear wave equations exterior to star-shaped regions in two space dimensions. The key novelty is that we establish a Morawetz type energy estimate for the perturbed inhomogeneous wave equation in the exterior domain, which yields $t^{-\frac12}$ decay inside the cone. In addition, two new weighted $L^2$ product estimates are established to produce $t^{-\frac12}$ decay close to the cone. We then show that the existence lifespan $T_\e$ for the quasilinear wave equations with general quadratic nonlinearity satisfies \begin{equation*} \varepsilon^2T_{\varepsilon}\ln^3T_{\varepsilon}=A, \end{equation*} for some fixed positive constant $A$, which is almost sharp (with some logarithmic loss) comparing to the known result of the corresponding Cauchy problem.