On the fractional heat semigroup and product estimates in Besov spaces and applications in theoretical analysis of the fractional Keller-Segel system

TL;DR

This paper studies the fractional Keller-Segel system using advanced Besov space estimates, establishing global existence, self-similarity, and uniqueness of solutions without smallness constraints.

Contribution

It introduces new bilinear estimates for fractional heat semigroups in Besov spaces and applies them to prove global existence and uniqueness of solutions for the fractional Keller-Segel system.

Findings

Established decay and integral estimates for Mittag-Leffler operators in Besov spaces.

Proved existence of global solutions and self-similar solutions in critical Besov spaces.

Achieved uniqueness of solutions without small initial data assumptions.

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 analyze the fractional heat semigroup obtaining time decay and integral estimates of the Mittag-Leffler operators in critical Besov spaces, and prove a bilinear estimate derived from the nonlinearity of the Keller-Segel system, without using auxiliary norms. We use these results in order to prove the existence of global solutions in critical homogeneous Besov spaces employing only the norm of the natural persistence space, 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.