AbstractIn this paper, we study the blow-up and global existence of solutions to the following time fractional nonlinear diffusion equations$$\begin{array}{} \displaystyle \left\{\begin{array}{l}{_0^CD_t^\alpha u}-\triangle u={_0I_t^{1-\gamma}}(|u|^{p-1}u), \, \, x\in \mathbb{R}^N,\, \, t\gt 0,\\ u(0,x)=u_0(x),\, \, x\in \mathbb{R}^N, \end{array}\right. \end{array}$$where 0 <α<γ< 1,p> 1,u0∈C0(ℝN),$\begin{array}{} {_0I_t^{\theta}} \end{array}$denotes left Riemann-Liouville fractional integrals of orderθ.$\begin{array}{} {_0^CD_t^\alpha u}=\frac{\partial}{\partial t}{_0I_t^{1-\alpha}} \end{array}$(u(t,x) −u(0,x))}. Letβ= 1 −γ. We prove that if 1 <p<p∗=$\begin{array}{} \max\big\{1+\frac{\beta}{\alpha},1+\frac{2(\alpha+\beta)}{\alpha N}\big\} \end{array}$, the solutions of (1.1) blows up in a finite time. IfN<$\begin{array}{} \frac{2(\alpha+\beta)}{\beta} \end{array}$,p≥p∗orN≥$\begin{array}{} \frac{2(\alpha+\beta)}{\beta} \end{array}$,p>p∗, and ∥u0∥Lqc(ℝN)is sufficiently small, where$\begin{array}{} q_c=\frac{N\alpha(p-1)}{2(\alpha+\beta)} \end{array}$, the solutions of (1.1) exists globally.
Blow-up and global existence for a kinetic equation of swarm formation
