Global small data weak solutions of 2-D semilinear wave equations with scale-invariant damping, III

2025/07/11 by Qianqian Li, Li, Qianqian, Huicheng Yin +1
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2507.08274

openalex publication_date 2025/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For the 2-D semilinear wave equation with scale-invariant damping \square u+\fracμt∂_tu=|u|^p, where t≥ 1, μ>0 and p>1, it is conjectured that the global small data weak solution u exists when p>p_s(2+μ) =(μ+3+√(μ²+14μ+17))/(2(μ+1)) for 0<μ≤ 2 and p>p_f(2)=2 for μ≥ 2. In our previous papers, the global small solution u has been obtained for p>p_s(2+μ) and 0<μ<2 but μ\not=1. In the present paper, by the vector field method together with the delicate analysis on the Bessel functions, we will show the global existence of small solution u for p>2 and μ>2. In forthcoming paper, for μ=1 and p>p_s(2+μ)=p_s(3)=1+√ 2, the global solution u is also obtained. Therefore, collecting our series of conclusions together with partial results from others, this open question has been solved completely.

Citations

• Global small data weak solutions of 2-D semilinear wave equations with scale-invariant damping, II
• Global small data weak solutions of 2-D semilinear wave equations with scale-invariant damping, I
• On global smooth solutions to the 2D isentropic and irrotational Chaplygin gases with short pulse data
• Life-Span of Semilinear Wave Equations with Scale-invariant Damping: Critical Strauss Exponent Case
• Life-span of solutions to semilinear wave equation with time-dependent critical damping for specially localized initial data
• Global solutions of quasilinear wave equations
• Solution Representations for a Wave Equation with Weak Dissipation
• The null condition for quasilinear wave equations in two space dimensions I
• Uniform decay estimates and the lorentz invariance of the classical wave equation

Related

• Global small data weak solutions of 2-D semilinear wave equations with scale-invariant damping, II
• Global small data weak solutions of 2-D semilinear wave equations with scale-invariant damping, I
• Global existence of small data weak solutions to the semilinear wave equations with time-dependent scale-invariant damping
• Global existence for small amplitude semilinear wave equations with time-dependent scale-invariant damping
• Global existence for semilinear wave equations with scaling invariant damping in 3-D