Superlinear transmission in an indirect signal production chemotaxis system

TL;DR

This paper investigates a chemotaxis system with nonlinear signal transmission, establishing conditions under which solutions remain globally bounded using Sobolev regularity techniques.

Contribution

It proves global boundedness of solutions for a chemotaxis model with superlinear transmission under new parameter conditions.

Findings

Solutions are globally bounded if 0<α<min{4/n, 1+2/n}.

Maximal Sobolev regularity is used to establish boundedness.

The results extend known bounds to superlinear transmission cases.

Abstract

In this paper, the indirect signal production system with nonlinear transmission is considered \[ \left\{ \begin{array}{lll} & u_t = \Delta u-\nabla\cdot(u \nabla v), \\ \displaystyle & v_t =\Delta v-v+w,\\ \displaystyle & w_t =\Delta w-w+ f(u) \end{array} \right. \] in a bounded smooth domain $\Omega\subset \mathbb{R}^n$ associated with homogenous Neumann boundary conditions, where $f\in C^1([0,\infty))$ satisfies $0\le f(s) \le s^{\alpha}$ with $\alpha>0$. It is known that the system possesses a global bounded solution if $0<\alpha<\frac 4n$ when $n\ge 4$. In the case $n\le 3$ and if we consider superlinear transmission, no regularity of $w$ or $v$ can be derived directly. In this work, we show that if $0<\alpha< \min\{\frac 4n,1+\frac 2n\}$, the solution is global and bounded via an approach based on the maximal Sobolev regularity.