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Suppression of blow up by a logistic source in 2D Keller-Segel system with fractional dissipation. (English) Zbl 1401.35312

The authors consider a two dimensional parabolic-elliptic Keller-Segel equation with a fractional diffusion of order \(\alpha \in(0, 2)\) and a logistic term. The existence of global in time regular solutions emanating from initial data with no size restrictions for some \(c < \alpha < 2\), where \(c \in(0, 2)\) depends on the equation’s parameters, is proved. For an even wider range of \(\alpha^\prime s\), the existence of global in time weak solutions for general initial data is also shown.

MSC:

35R11 Fractional partial differential equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B44 Blow-up in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35S10 Initial value problems for PDEs with pseudodifferential operators
92C17 Cell movement (chemotaxis, etc.)
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