Semilinear hyperbolic systems violating the null condition

2012/06/01 by Soichiro Katayama, Katayama, Soichiro, Toshiaki Matoba +3 · 1 citation
Engineering · Mathematics · Physics and Astronomy · #35B40 #35L71 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons #Stability and Controllability of Differential Equations #math.AP #msc:35B40 #msc:35L71

paper · pdf · doi:10.48550/arxiv.1206.0066

32 pages. Statements of main theorems have been changed in order to treat more general situation. Accordingly, the title and the abstarct have also been modified

openalex publication_date 2012/06/01 · arxiv created 2013/04/10 · arxiv updated 2013/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider systems of semilinear wave equations in three space dimensions with quadratic nonlinear terms not satisfying the null condition. We prove small data global existence of the classical solution under a new structural condition related to the weak null condition. For two-component systems satisfying this condition, we also observe a new kind of asymptotic behavior: Only one component is dissipated and the other one behaves like a free solution in the large time.

Citations

• A remark on decay rates of solutions for a system of quadratic nonlinear Schrödinger equations in 2D

Cited by

• Energy decay for systems of semilinear wave equations with dissipative structure in two space dimensions

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