Chemotaxis model with subcritical exponent in nonlocal reaction

TL;DR

This paper analyzes a chemotaxis model with nonlocal reactions, establishing conditions for global bounded solutions and comparing it to classical models to prevent blow-up without initial data restrictions.

Contribution

It introduces a chemotaxis system with nonlocal source terms, identifying regimes where solutions remain globally bounded, extending classical models.

Findings

Existence of unique global solutions under certain conditions

Energy estimates prevent blow-up without initial data restrictions

Comparison with classical Keller-Segel model highlights differences

Abstract

This paper deals with a parabolic-elliptic chemotaxis system with nonlocal type of source in the whole space. It's proved that the initial value problem possesses a unique global solution which is uniformly bounded. Here we identify the exponents regimes of nonlinear reaction and aggregation in such a way that their scaling and the diffusion term coincide (see Introduction). Comparing to the classical KS model (without the source term), it's shown that how energy estimates give natural conditions on the nonlinearities implying the absence of blow-up for the solution without any restriction on the initial data.