2024/12/12 by Ikeda, Masahiro, Zhao, Jinhong
#35B40(Secondary) #35L05 #35R11 (Primary) 35B44 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2412.09155
This article deals with the behavior in time of the solution to the Cauchy problem for a fractional wave equation with a weighted L1 initial data. Initially, we establish the global existence of the solution using Fourier methods and provide upper bounds for the L2 norm and the Hs norm of the solution for any dimension n∈ ℕ and s∈ (0,1). However, when n=1 and s ∈ [(1)/(2),1), %we have to assume that the initial velocity satisfies we have to impose a stronger assumption ∫ℝu1(x)dx=0. To remove this stronger assumption, we further use the Fourier splitting method, which yields the optimal blow-up rate for the L2 norm of the solutions. Specifically, when n=1, the optimal blow-up rate is t1-(1)/(2s) for s ∈ ((1)/(2),1) and √(log t) for s = (1)/(2).