## Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients.(English)Zbl 1478.35218

Summary: 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.

### MSC:

 35R11 Fractional partial differential equations 26A33 Fractional derivatives and integrals 35K15 Initial value problems for second-order parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35B44 Blow-up in context of PDEs 33E12 Mittag-Leffler functions and generalizations 44A20 Integral transforms of special functions
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