Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source

TL;DR

This paper proves the global boundedness and existence of classical solutions for a quasilinear Keller-Segel system with logistic source under certain conditions on the parameters, extending understanding of chemotaxis models.

Contribution

It establishes new boundedness criteria for solutions of a quasilinear Keller-Segel system with logistic growth, considering variable diffusion and sensitivity functions.

Findings

Global classical solutions exist under specified parameter conditions.

Solutions remain uniformly bounded over time.

The results extend previous boundedness criteria to more general nonlinearities.

Abstract

This paper deals with the Neumann boundary value problem for the system $$u_t=\nabla\cdot\left(D(u)\nabla u\right)-\nabla\cdot\left(S(u)\nabla v\right)+f(u) ,\quad x\in\Omega,\ t>0$$ $$v_t=\Delta v-v+u,\quad x\in\Omega,\ t>0$$ in a smooth bounded domain $\Omega\subset\mathbb{R}^n$ $(n\geq1)$, where the functions $D(u)$ and $S(u)$ are supposed to be smooth satisfying $D(u)\geq Mu^{-\alpha}$ and $S(u)\leq Mu^{\beta}$ with $M>0$, $\alpha\in\mathbb{R}$ and $\beta\in\mathbb{R}$ for all $u\geq1$, and the logistic source $f(u)$ is smooth fulfilling $f(0)\geq0$ as well as $f(u)\leq a-\mu u^{\gamma}$ with $a\geq0$, $\mu>0$ and $\gamma\geq1$ for all $u\geq0$. It is shown that if $\alpha+2\beta<\gamma-1+\frac{2}{n}$, for $1\leq\gamma<2$ and $\alpha+2\beta<\gamma-1+\frac{4}{n+2}$, for $\gamma\geq2$, then for sufficiently smooth initial data the problem possesses a unique global classical solution which is uniformly bounded.