Optimal $L^2$-blowup estimates of the Fractional Wave Equation

Abstract

This article deals with the behavior in time of the solution to the Cauchy problem for a fractional wave equation with a weighted $L^1$ initial data. Initially, we establish the global existence of the solution using Fourier methods and provide upper bounds for the $L^2$ norm and the $H^s$ norm of the solution for any dimension $n\in \mathbb{N}$ and $s\in (0,1)$. However, when $n=1$ and $s \in [\frac{1}{2},1)$, %we have to assume that the initial velocity satisfies we have to impose a stronger assumption $\int_{\mathbb{R}}u_1(x)dx=0$. To remove this stronger assumption, we further use the Fourier splitting method, which yields the optimal blow-up rate for the $L^2$ norm of the solutions. Specifically, when $n=1$, the optimal blow-up rate is $t^{1-\frac{1}{2s}}$ for $s \in (\frac{1}{2},1)$ and $\sqrt{\log t}$ for $s = \frac{1}{2}$.