Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source

TL;DR

This paper establishes the existence of global weak solutions and eventual smoothness for a 3D chemotaxis system with logistic growth, analyzing long-term behavior including decay and boundedness under certain conditions.

Contribution

It proves global weak solutions and eventual classical smoothness in 3D chemotaxis systems with logistic sources, extending understanding of solution regularity and asymptotics.

Findings

Existence of global weak solutions in 3D domain.

Solutions become classical after some time under certain parameters.

Decay and boundedness of solutions depending on parameter .

Abstract

We prove existence of global weak solutions to the chemotaxis system $ u_t=\Delta u - \nabla\cdot (u\nabla v) +\kappa u -\mu u^2 $ $ v_t=\Delta v-v+u $ under homogeneous Neumann boundary conditions in a smooth bounded convex domain $\Omega\subset R^n$, for arbitrarily small values of $\mu>0$. Additionally, we show that in the three-dimensional setting, after some time, these solutions become classical solutions, provided that $\kappa$ is not too large. In this case, we also consider their large-time behaviour: We prove decay if $\kappa\leq 0$ and the existence of an absorbing set if $\kappa>0$ is sufficiently small. Keywords: chemotaxis, logistic source, existence, weak solutions, eventual smoothness