On a two-species cross attraction system in higher dimensions

TL;DR

This paper studies a complex two-species chemotaxis model in higher dimensions, establishing conditions for local existence, and identifying parameters that determine whether solutions exist globally or blow up.

Contribution

It introduces a new analysis of a degenerate chemotaxis system with two stimuli in dimensions d ≥ 3, highlighting critical parameter values for solution behavior.

Findings

Existence of local-in-time solutions for any initial mass.

Identification of critical parameter values for global existence.

Conditions under which solutions blow up in finite time.

Abstract

We consider a degenerate chemotaxis model with two-species and two-stimuli in dimension $d \geq 3$. Under the hypothesis of integrable initial data with finite second moment and energy, we show local-in-time existence for any mass of free-energy solutions, namely weak solutions with some free energy estimates. We exhibit that the qualitative behavior of solutions is decided by a set of critical values: there is a critical value of a parameter pair in the system of equations for which there is a global-in-time energy solution and there exist blowing-up free-energy solutions under a criticality condition is violated for the parameter pair.