Blow-up and global existence of solutions for a time fractional diffusion equation

Abstract

In this paper, we study the blow-up and global existence of solutions to the following time fractional nonlinear diffusion equations

$$\left\{ {\begin{array}{*{20}{c}} {\mathop 0\limits^C D_t^\alpha u - \Delta u{ = _0}I_t^{1 - \gamma }\left( {{{\left| u \right|}^{p - 1}}u} \right),\;x \in {^N},\;t > 0,} \\ {u\left( {0,x} \right) = {u_0}\left( x \right),\;x \in {^N},} \\ \end{array}} \right.$$

where 0 < α < γ < 1, p > 1, u₀ ∈ C₀(ℝ^N), $_0I_t^\theta$ denotes left Riemann-Liouville fractional integrals of order θ. $\mathop 0\limits^C D_t^\alpha u = \frac{\partial }{{\partial t}}{}_0I_t^{1 - \alpha }\left( {u\left( {t,x} \right) - u\left( {0,x} \right)} \right)$. Let β = 1 − γ. We prove that if $1<p<p* = \max \left\{ {1 + \frac{\beta }{\alpha },1 + \frac{{2\left( {\alpha + \beta } \right)}}{{\alpha N}}} \right\}$, the solutions of (1.1) blows up in a finite time. If $N<\frac{{2\left( {\alpha + \beta } \right)}}{\beta },p \ge p*\;or\;N \ge \;\frac{{2\left( {\alpha + \beta } \right)}}{\beta },p > p*$, and ∥u₀∥Lq_c(ℝ^N) is sufficiently small, where ${q_c} = \frac{{N\alpha \left( {p - 1} \right)}}{{2\left( {\alpha + \beta } \right)}}$, the solutions of (1.1) exists globally.