Existence of global solutions to semilinear damped wave equations with nonlinearities of derivative type

2025/12/07 by Van Duong, Dinh, Dao, Tuan Anh
Engineering · Mathematics · #35A01 #35B33 #35L15 #35L71 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations

paper · doi:10.48550/arxiv.2512.06789

openalex publication_date 2025/12/07 · openalex created_date 2025/12/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we would like to consider the semi-linear damped wave equation with the power nonlinearity of derivative type |u_t|^p. The main contribution of this work is to improve the results for global (in time) solution existence in a comparison with the pioneering paper \citeMatsumura1976 of Matsumura, who first established that the solutions exist globally for p > 1 (n = 1) and p ≥ 2 (n ≥ 2). More precisely, we have extended such a result for any p > 1 (n = 1,2) and p > 3/2 (n = 3). Our approach relies on constructing appropriately weighted solution spaces linked to the delicate application of several tools from Harmonic Analysis and Banach fixed-point theorem.

Citations

• Lifespan estimates for semilinear damped wave equation in a two-dimensional exterior domain
• 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

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