Existence and uniqueness of global solutions for time fractional Keller-Segel system and fractional dissipation

Abstract

This paper is concerned with the fractional Keller-Segel system in the temporal and spatial variables. We consider fractional dissipation for the physical variables including a fractional dissipation mechanism for the chemotactic diffusion, as well as a time fractional variation assumed in the Caputo sense. We prove new results concerning to the existence of global solutions in critical homogeneous Besov spaces without using auxiliary norms, including the existence of self-similar solutions, which constitutes a persistence result in this framework. In addition, we prove a uniqueness result without assuming any smallness condition of the initial data.