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arXiv · 2507.06897

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

Lai Ning-An·Ren Cui·Xu Wei
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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.

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Lai Ning-An, Ren Cui, Xu Wei. 2025-07-09. Long-Time Existence of Quasilinear Wave Equations Exterior to Star-shaped Obstacle in $2\mathbf{D}$. https://arxiv.org/abs/2507.06897

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