Abstract
This article deals with a semilinear time-space fractional diffusion equation and a coupled semilinear diffusion system of two time-space fractional diffusion equations in the whole Euclidean space. The time-fractional derivative used here is in the sense of \(\psi \)-Caputo derivative with order in (0, 1), and the space derivative is the so-called fractional Laplacian with order also in (0, 1). The Cauchy problem of the semilinear equation is firstly studied by constructing its mild solution and weak solution where its mild solution is shown to be also its weak solution. The finite-time blow-up and global existence of the solution to this semilinear equation are studied. Following almost the same procedure, the blow-up in a finite time and global existence of the solution to the coupled semilinear system are investigated too. Finally, the finite-time blow-up is tested by numerical simulations.
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The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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Communicated by Anthony Bloch.
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The work is partially supported by the National Natural Science Foundation of China (Nos. 11872234 and 11926319) and Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (No. 2021L573)
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Li, C., Li, Z. The Finite-Time Blow-Up for Semilinear Fractional Diffusion Equations with Time \(\psi \)-Caputo Derivative. J Nonlinear Sci 32, 82 (2022). https://doi.org/10.1007/s00332-022-09841-6
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DOI: https://doi.org/10.1007/s00332-022-09841-6
Keywords
- Fractional partial differential equation
- \(\psi \)-Caputo derivative
- Fractional Laplacian
- Blow-up
- Mild solution
- Weak solution
- Global solution