Global solvability and boundedness in the $N$-dimensional quasilinear chemotaxis model with logistic source and consumption of chemoattractant

TL;DR

This paper establishes conditions for global existence and boundedness of solutions in a complex chemotaxis model with logistic growth, extending previous results to broader parameter ranges and initial data.

Contribution

It provides new criteria ensuring global bounded solutions for the chemotaxis system with nonlinear diffusion and logistic source, generalizing earlier work to higher dimensions and more general diffusion functions.

Findings

Global bounded solutions exist under specified conditions.

Extends previous results to higher dimensions and broader diffusion functions.

Provides explicit bounds depending on initial data and parameters.

Abstract

We consider the following chemotaxis model %fully parabolic Keller-Segel system with logistic source $$ \left\{\begin{array}{ll} u_t=\nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)+\mu (u-u^2),\quad x\in \Omega, t>0, \disp{v_t-\Delta v=-uv },\quad x\in \Omega, t>0, %\disp{\tau w_t+\delta w=u },\quad %x\in \Omega, t>0, \disp{(\nabla D(u)-\chi u\cdot \nabla v)\cdot \nu=\frac{\partial v}{\partial\nu}=0},\quad x\in \partial\Omega, t>0, \disp{u(x,0)=u_0(x)},\quad v(x,0)=v_0(x),~~ x\in \Omega \end{array}\right. $$ on a bounded domain $\Omega\subset\mathbb{R}^N(N\geq1)$, with smooth boundary $\partial\Omega, \chi$ and $\mu$ are positive constants. Besides appropriate smoothness assumptions, in this paper it is only required that $D(u)\geq C_{D}(u+1)^{m-1}$ for all $u\geq 0$ with some $C_{D} > 0$ and some $$ m>\left\{\begin{array}{ll} 1-\frac{\mu}{\chi[1+\lambda_{0}\|v_0\|_{L^\infty(\Omega)}2^{3}]}~~\mbox{if}~~ N\leq2, % >1+\frac{(N+2-2r)^+}{N+2}~~~~~~\mbox{if}~~ % \frac{N+2}{2}\geq r\geq\frac{N+2}{N}, 1~~~~~~\mbox{if}~~ N\geq3, \end{array}\right. $$ then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded, where $\lambda_{0}$ is a positive constant which is corresponding to the maximal sobolev regularity. The results of this paper extends the results of Jin (J. Diff. Eqns., 263(9)(2017), 5759-5772), who proved the possibility of boundness of weak solutions, in the case $m>1$ and $N=3$.