A new result for global existence and boundedness of solutions to a parabolic--parabolic Keller--Segel system with logistic source

TL;DR

This paper establishes conditions for the global existence, boundedness, and decay of solutions to a parabolic Keller--Segel system with logistic source, advancing understanding of chemotaxis models with logistic growth.

Contribution

It provides new criteria ensuring global weak and classical solutions, including boundedness and decay, for the Keller--Segel system with logistic source, depending on parameters.

Findings

Global weak solutions exist if >0.

Global bounded classical solutions exist under certain parameter conditions.

Solutions decay to zero in the -infinity norm as time approaches infinity.

Abstract

We consider the following fully parabolic Keller--Segel system with logistic source $$ \left\{\begin{array}{ll} u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+ au-\mu u^2,\quad x\in \Omega, t>0, \disp{v_t=\Delta v- v +u},\quad x\in \Omega, t>0, \end{array}\right.\eqno(KS) $$ over a bounded domain $\Omega\subset\mathbb{R}^N(N\geq1)$, with smooth boundary $\partial\Omega$, the parameters $a\in \mathbb{R}, \mu>0, \chi>0$. It is proved that if $\mu>0$, then $(KS)$ admits a global weak solution, while if $\mu>\frac{(N-2)_{+}}{N}\chi C^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1}$, then $(KS)$ possesses a global classical solution which is bounded, where $C^{\frac{1}{\frac{N} {2}+1}}_{\frac{N}{2}+1}$ is a positive constant which is corresponding to the maximal Sobolev regularity. Apart from this, we also show that if $a = 0$ and $\mu>\frac{(N-2)_{+}}{N}\chi C^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1}$, then both $u(\cdot, t)$ and $v(\cdot, t)$ decay to zero with respect to the norm in $L^\infty(\Omega)$ as $t\rightarrow\infty$.