Global behaviors of defocusing semilinear wave equations

2019/08/01 by Shiwu Yang, Yang, Shiwu
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Navier-Stokes equation solutions #math.AP

paper · pdf · doi:10.48550/arxiv.1908.00606

arxiv created 2019/08/01 · openalex publication_date 2019/08/01 · arxiv updated 2019/08/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we investigate the global behaviors of solutions to defocusing semilinear wave equations in ℝ^1+d with d≥ 3. We prove that in the energy space the solution verifies the integrated local energy decay estimates for the full range of energy subcritical and critical power. For the case when p>1+(2)/(d-1), we derive a uniform weighted energy bound for the solution as well as inverse polynomial decay of the energy flux through hypersurfaces away from the light cone. As a consequence, the solution scatters in the energy space and in the critical Sobolev space for p with an improved lower bound. This in particular extends the existing scattering results to higher dimensions without spherical symmetry.

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