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Global existence, local existence and blow-up of mild solutions for abstract time-space fractional diffusion equations. (English) Zbl 1523.35283

Summary: In this paper, we consider initial boundary value problems for abstract fractional diffusion equations \(\partial_t^{\beta}u+(-\Delta)^su=g(t,x,u)\) 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.

MSC:

35R11 Fractional partial differential equations
26A33 Fractional derivatives and integrals
35B44 Blow-up in context of PDEs
35K15 Initial value problems for second-order parabolic equations
35K90 Abstract parabolic equations
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