Boundedness and finite-time blow-up in a quasilinear parabolic-elliptic chemotaxis system with logistic source and nonlinear production

TL;DR

This paper investigates conditions under which solutions to a complex chemotaxis system with nonlinear production and logistic source are bounded or blow up in finite time, extending previous results to more general functions.

Contribution

It establishes new criteria for boundedness and finite-time blow-up in a generalized chemotaxis model with nonlinear diffusion, chemotactic sensitivity, and production functions.

Findings

Derived conditions for solution boundedness.

Identified parameters leading to finite-time blow-up.

Extended previous results to more general nonlinear functions.

Abstract

This paper deals with the quasilinear parabolic-elliptic chemotaxis system with logistic source and nonlinear production, \begin{equation*} \begin{cases} u_t=\nabla \cdot (D(u) \nabla u) - \nabla \cdot (S(u)\nabla v) + \lambda u - \mu u^{\kappa}, & x\in\Omega,\ t>0, \\[1mm] 0=\Delta v - \overline{M_f}(t) + f(u), & x\in\Omega,\ t>0, \end{cases} \end{equation*} where $\lambda>0$, $\mu>0$, $\kappa>1$ and $\overline{M_f}(t):=\frac{1}{|\Omega|}\int_{\Omega} f(u(x,t))\,dx$, and $D$, $S$ and $f$ are functions generalizing the prototypes \begin{align*} D(u)=(u+1)^{m-1},\quad S(u)=u(u+1)^{\alpha-1}\quad\mbox{and}\quad f(u)=u^\ell \end{align*} with $m\in\mathbb{R}$, $\alpha>0$ and $\ell>0$. In the case $m=\alpha=\ell=1$, Fuest (NoDEA Nonlinear Differential Equations Appl.; 2021; 28; 16) obtained conditions for $\kappa$ such that solutions blow up in finite time. However, in the above system boundedness and finite-time blow-up of solutions have been not yet established. This paper gives boundedness and finite-time blow-up under some conditions for $m$, $\alpha$, $\kappa$ and $\ell$.