On the fractional doubly parabolic Keller-Segel system modelling chemotaxis.

This work is concerned with the time-fractional doubly parabolic Keller-Segel system in ℝ^N (N ≽ 1), and we derive some refined results on the large time behavior of solutions which are presupposed to enjoy some uniform boundedness properties. Moreover, the well-posedness and the asymptotic stability of solutions in Marcinkiewicz spaces are studied. The results are achieved by means of an appropriate estimation of the system nonlinearities in the course of an analysis based on Duhamel-type representation formulae and the Kato-Fujita framework which consists in constructing a fixed-point argument by using a suitable time-dependent space. [ABSTRACT FROM AUTHOR]