2024/05/14 by Daoyin He, Qianqian Li, He, Daoyin +3
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2405.08407
openalex publication_date 2024/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we are concerned with the global existence of small data weak solutions to the n-dimensional semilinear wave equation ∂t2u-Δu+\fracμt∂tu=|u|p with time-dependent scale-invariant damping, where n≥ 2, t≥ 1, μ∈(0,1)∪(1,2] and p>1. This equation can be changed into the semilinear generalized Tricomi equation ∂t2u-tmΔu=tα(m)|u|p, where m=m(μ)>0 and α(m)∈\Bbb R are two suitable constants. At first, for the more general semilinear Tricomi equation ∂t2v-tmΔv=tα|v|p with any fixed constant m>0 and arbitrary parameter α∈\Bbb R, we shall show that in the case of α≤ -2, n≥ 3 and p>1, the small data weak solution v exists globally; in the case of α>-2, through determining the conformal exponent pconf(n,m,α)>1, the global small data weak solution v exists when some extra restrictions of p≥ pconf(n,m,α) are given. Returning to the original equation ∂t2u-Δu+\fracμt∂tu=|u|p, the corresponding global existence results on the small data solution u can be obtained.