Life span of blowing-up solutions to the Cauchy problem for a time–space fractional diffusion equation

Abstract

In this paper, we study the blow-up, and global existence of solutions to the following time–space fractional diffusion problem 0CDtαu(x,t)+(−Δ)β∕2u(x,t)=0Jtγ|u|p−1u(x,t),x∈RN,t>0,u(x,0)=u0(x)∈C0(RN),where 0<α<1−γ, 0<γ<1, 0<β≤2, p>1, 0Jtγ denotes the left Riemann–Liouville fractional integral of order γ, 0CDtα is the Caputo fractional derivative of order α and (−Δ)β∕2 stands for the fractional Laplacian operator of order β∕2. We show that if p<1+β(α+γ)∕αN, then every nonnegative solution blows up in finite time, and if p≥1+β(α+γ)∕αN and ‖u0‖Lqc(RN) is sufficiently small, where qc=αN(p−1)∕β(α+γ), then the problem has global solutions. Finally, we give an upper bound estimate of the life span of blowing-up solutions.