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

In this paper, we investigate the blow-up and global existence of solutions to the following time fractional nonlinear diffusion equations {_0^C D_t^\alpha u}-\triangle u=|u|^{p-1}u, x\in \Bbb{R}^N,\ t\ge0, u(0,x)=u_0(x), x\in \Bbb{R}^N, where $0\le\alpha\le 1$, $p\ge 1$, $u_0\in C_0(\Bbb{R}^N)$ and ${_0^CD_t^\alpha u}=({\partial}/{\partial t}){_0^{}I_t^{1-\alpha}(u(t,x)-u_0(x))}$, ${_0^{}I_t^{1-\alpha}}$ denotes left Riemann--Liouville fractional integrals of order $1-\alpha$. We prove that if $1\le p\le 1+{2}/{N}$, then every nontrivial nonnegative solution blow-up in finite time, and if $p\geq 1+{2}/{N}$ and $\|u_0\|_{L^{q_c}(\Bbb{R}^N)}$, $q_c={N(p-1)}/{2}$ is sufficiently small, then the problem has global solution