Boundedness of large-time solutions to a chemotaxis model with nonlocal and semilinear flux

TL;DR

This paper investigates the boundedness of solutions to a one-dimensional chemotaxis model with nonlocal and semilinear flux, establishing conditions under which solutions remain bounded for large times, even with weaker diffusions and large initial data.

Contribution

It provides new boundedness results for a semilinear Keller-Segel system with critical nonlocal diffusion, including cases with weaker diffusions and large initial data, under explicit conditions.

Findings

Boundedness of solutions under general semilinearity conditions

Extension to weaker diffusions with logistic damping

Boundedness for arbitrarily large initial data when r > 1

Abstract

A semilinear version of parabolic-elliptic Keller-Segel system with the \emph{critical} nonlocal diffusion is considered in one space dimension. We show boundedness of weak solutions under very general conditions on our semilinearity. It can degenerate, but has to provide a stronger dissipation for large values of a solution than in the critical linear case or we need to assume certain (explicit) data smallness. Moreover, when one considers a logistic term with a parameter $r$, we obtain our results even for diffusions slightly weaker than the critical linear one and for arbitrarily large initial datum, provided $r >1$. For a mild logistic dampening, we can improve the smallness condition on the initial datum up to $\sim \frac{1}{1-r}$.