## Topological Methods in Nonlinear Analysis

### 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 of solutions to the following time fractional nonlinear diffusion equations \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 ${_0^CD_t^\alpha u}=({\partial}/{\partial t}){_0^{}I_t^{1-\alpha}(u(t,x)-u_0(x))}$, ${_0^{}I_t^{1-\alpha}}$ denotes left Riemann-Liouville fractional integrals of order $1-\alpha$. We prove that if $1 < p < 1+2/{N}$, then every nontrivial nonnegative solution blow-up in finite time, and if $p\geq 1+2/{N}$ and $\|u_0\|_{L^{q_c}(\mathbb{R}^N)}$, $q_c=N(p-1)/{2}$ is sufficiently small, then the problem has global solution.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 46, Number 1 (2015), 69-92.

Dates
First available in Project Euclid: 30 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1459343886

Digital Object Identifier
doi:10.12775/TMNA.2015.038

Mathematical Reviews number (MathSciNet)
MR3443679

Zentralblatt MATH identifier
1362.35325

#### Citation

Zhang, Quan-Guo; Sun, Hong-Rui. The blow-up and global existence of solutions of Cauchy problems for a time fractional diffusion equation. Topol. Methods Nonlinear Anal. 46 (2015), no. 1, 69--92. doi:10.12775/TMNA.2015.038. https://projecteuclid.org/euclid.tmna/1459343886

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