Long-Time Existence of Quasilinear Wave Equations Exterior to Star-shaped Obstacle in 2D

2025/07/09 by N. Lai, Ning-An, Lai, Ren Hao Cui +3
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2507.06897

openalex publication_date 2025/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 L² 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 ε²T_εln³T_ε=A, for some fixed positive constant A, which is almost sharp (with some logarithmic loss) comparing to the known result of the corresponding Cauchy problem.

Citations

• Global smooth solutions of 2-D quadratic quasilinear wave equations with null conditions in exterior domains
• Almost Global Existence for 4-Dimensional Quasilinear Wave Equations in Exterior Domains
• Almost global existence for exterior Neumann problems of semilinear wave equations in 2D
• The lifespan for 3-dimensional quasilinear wave equations in exterior domains
• Almost global existence for nonlinear wave equations in an exterior domain in two space dimensions
• Global existence of null-form wave equations in exterior domains
• Elastic waves in exterior domains, Part II: Global existence with a null structure
• Global existence of solutions to multiple speed systems of quasilinear wave equations in exterior domains
• Global existence for Dirichlet-wave equations with quadratic nonlinearties in high dimensions
• Almost global existence for quasilinear wave equations in three space dimensions
• The null condition for quasilinear wave equations in two space dimensions I
• Almost global existence for some semilinear wave equations
• Global existence for a quasilinear wave equation outside of star-shaped domains
• Uniform decay estimates and the lorentz invariance of the classical wave equation

Related

• The lifespan for 3-dimensional quasilinear wave equations in exterior domains
• Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles
• Almost Global Existence for 4-Dimensional Quasilinear Wave Equations in Exterior Domains
• Global existence for a quasilinear wave equation outside of star-shaped domains
• Long-time existence for systems of quasilinear wave equations