Global existence for small amplitude semilinear wave equations with time-dependent scale-invariant damping

2025/01/03 by Daoyin He, He, Daoyin, Sun Ya-qing +3 · 2 citations
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods for differential equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2501.01670

openalex publication_date 2025/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we prove a sharp global existence result for semilinear wave equations with time-dependent scale-invariant damping terms if the initial data is small. More specifically, we consider Cauchy problem of ∂_t²u-Δu+\fracμt∂_tu=|u|^p, where n≥ 3, t≥ 1 and μ∈(0,1)∪(1,2). For critical exponent p_crit(n,μ) which is the positive root of (n+μ-1)p²-(n+μ+1)p-2=0 and conformal exponent p_conf(n,μ)=(n+μ+3)/(n+μ-1), we establish global existence for n≥3 and p_crit(n,μ)0 and α(m)∈\Bbb R are two suitable constants, then we investigate more general semilinear Tricomi equation ∂_t²v-t^mΔv=t^α|v|^p and establish related weighted Strichartz estimates. Returning to the original wave equation, the corresponding global existence results on the small data solution u can be obtained.

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• Boundedness and decay of waves on spatially flat decelerated FLRW spacetimes
• Morawetz type estimate for damped wave equation in ℝⁿ (n≥ 4) and its application

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