The blow-up and global existence of solutions of Cauchy problems for a time fractional diffusion equation

Abstract

In this paper, we investigate the blow-up and global existence ofsolutions to the following time fractional nonlinear diffusionequations\begin{equation*}\begin{cases}{_0^C D_t^\alpha u}-\triangle u=|u|^{p-1}u, & x\in \mathbb{R}^N,\ t > 0,\\u(0,x)=u_0(x), & x\in \mathbb{R}^N,\end{cases}\end{equation*}where $0 < \alpha < 1$, $p > 1$, $u_0\in C_0(\mathbb{R}^N)$ and${_0^CD_t^\alpha u}=({\partial}/{\partialt}){_0^{}I_t^{1-\alpha}(u(t,x)-u_0(x))}$, ${_0^{}I_t^{1-\alpha}}$denotes left Riemann-Liouville fractional integrals of order$1-\alpha$. We prove that if $1 < p < 1+2/{N}$, then everynontrivial nonnegative solution blow-up in finite time, and if$p\geq 1+2/{N}$ and $\|u_0\|_{L^{q_c}(\mathbb{R}^N)}$,$q_c=N(p-1)/{2}$ is sufficiently small, then the problem hasglobal solution.