Global existence, local existence and blow-up of mild solutions for abstract time-space fractional diffusion equations

Abstract

In this paper, we consider initial boundary value problems for abstract fractional diffusion equations $\partial_{t}^{\beta}u+(-\Delta)^{s}u=g(t,x,u)$ with the Caputo time fractional derivatives and fractional Laplacian operators. When $g(t,x,u)$ satisfies condition (G), problems can be applied by a strong maximum principle involving time-space fractional derivatives. Hence, we establish the global existence and uniqueness of mild solution by upper and lower solutions method. Moreover, the mild solution mentioned above turns out to be a classical solution. When condition (G) does not hold, then we study nonexistence of global solutions under certain conditions, and we obtain the local existence and blow-up of mild solutions. Further, we conclude that the first eigenvalue $\lambda_1$ seems to be a critical value for nonlinear problems.