### Electronic Research Archive

2021, Issue 6: 3581-3607. doi: 10.3934/era.2021052

# Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients

• Received: 01 March 2021 Revised: 01 May 2021 Published: 22 July 2021
• Primary: 26A33, 35K15; Secondary: 35B40, 33E12, 44A20

• The semi-linear problem of a fractional diffusion equation with the Caputo-like counterpart of a hyper-Bessel differential is considered. The results on existence, uniqueness and regularity estimates (local well-posedness) of the solutions are established in the case of linear source and the source functions that satisfy the globally Lipschitz conditions. Moreover, we prove that the problem exists a unique positive solution. In addition, the unique continuation of solutions and a finite-time blow-up are proposed with the reaction terms are logarithmic functions.

Citation: Vo Van Au, Jagdev Singh, Anh Tuan Nguyen. Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients[J]. Electronic Research Archive, 2021, 29(6): 3581-3607. doi: 10.3934/era.2021052

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• The semi-linear problem of a fractional diffusion equation with the Caputo-like counterpart of a hyper-Bessel differential is considered. The results on existence, uniqueness and regularity estimates (local well-posedness) of the solutions are established in the case of linear source and the source functions that satisfy the globally Lipschitz conditions. Moreover, we prove that the problem exists a unique positive solution. In addition, the unique continuation of solutions and a finite-time blow-up are proposed with the reaction terms are logarithmic functions.

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