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

Abstract

A semilinear version of parabolic-elliptic Keller-Segel system with the 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 $r \gt 1$. For a mild logistic dampening, we can improve the smallness condition on the initial datum up to $\sim {1}/({1-r})$.