Long-Time Existence of Quasilinear Wave Equations Exterior to Star-shaped Obstacle in $2\mathbf{D}$
Long-Time Existence of Quasilinear Wave Equations Exterior to Star-shaped Obstacle in $2\mathbf{D}$
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.
Lai Ning-An、Ren Cui、Xu Wei
数学物理学
Lai Ning-An,Ren Cui,Xu Wei.Long-Time Existence of Quasilinear Wave Equations Exterior to Star-shaped Obstacle in $2\mathbf{D}$[EB/OL].(2025-07-09)[2025-07-23].https://arxiv.org/abs/2507.06897.点此复制
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