TY - JOUR AU - Fu, Yongqiang AU - Zhang, Xiaoju PY - 2022/09/13 Y2 - 2024/03/29 TI - Global existence, local existence and blow-up of mild solutions for abstract time-space fractional diffusion equations JF - Topological Methods in Nonlinear Analysis JA - TMNA VL - 60 IS - 2 SE - DO - 10.12775/TMNA.2021.015 UR - https://apcz.umk.pl/TMNA/article/view/39985 SP - 415 - 440 AB - 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 theCaputo 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 $\lambda_1$ seemsto be a critical value for nonlinear problems. ER -