Critical curve for the weakly coupled system of damped wave equations with mixed nonlinearities

2025/06/19 by Van Duong, Dinh, Dao, Tuan Anh, Ikeda, Masahiro
Engineering · Mathematics · #35A01 #35B44 #35L51 #35L52 #35L71 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2506.16126

openalex publication_date 2025/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we would like to consider the Cauchy problem for a weakly coupled system of semi-linear damped wave equations with mixed nonlinear terms. Our main objective is to draw conclusions about the critical curve of this problem using tools from Harmonic Analysis. Precisely, we obtain a new critical curve pq = 1+ (2)/(n) for n =1,2 by proving global (in time) existence of small data Sobolev solutions when pq > 1 +(2)/(n) and blow-up of weak solutions in finite time even for small data when pq < 1+ (2)/(n) for n ≥ 1. From this, we infer the impact of the nonlinearities of time derivative-type on the critical curve associated with the system.

Citations

• Critical regularity of nonlinearities in semilinear classical damped\n wave equations
• An application of L¹ estimates for oscillating integrals to parabolic like semi-linear structurally damped σ-evolution models
• L² asymptotic profiles of solutions to linear damped wave equations
• The sharp lifespan estimate for semilinear damped wave equation with Fujita critical power in high dimensions

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