# Boundedness in a chemotaxis system with consumed chemoattractant and
produced chemorepellent

Research paper by **Silvia Frassu, Giuseppe Viglialoro**

Indexed on: **25 Sep '20**Published on: **24 Sep '20**Published in: **arXiv - Mathematics - Analysis of PDEs**

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#### Abstract

We study this zero-flux attraction-repulsion chemotaxis model, with linear
and superlinear production $g$ for the chemorepellent and sublinear rate $f$
for the chemoattractant: \begin{equation}\label{problem_abstract}
\tag{$\Diamond$} \begin{cases} u_t= \Delta u - \chi \nabla \cdot (u \nabla
v)+\xi \nabla \cdot (u \nabla w) & \text{ in } \Omega \times (0,T_{max}),\\
v_t=\Delta v-f(u)v & \text{ in } \Omega \times (0,T_{max}),\\ 0= \Delta w -
\delta w + g(u)& \text{ in } \Omega \times (0,T_{max}). %u(x,0)=u_0(x), \;
v(x,0)=v_0(x) & x \in \bar\Omega. \end{cases} \end{equation} In this problem,
$\Omega$ is a bounded and smooth domain of $\R^n$, for $n\geq 1$,
$\chi,\xi,\delta>0$, $f(u)$ and $g(u)$ reasonably regular functions
generalizing the prototypes $f(u)=K u^\alpha$ and $g(u)=\gamma u^l$, with
$K,\gamma>0$ and proper $ \alpha, l>0$. Once it is indicated that any
sufficiently smooth $u(x,0)=u_0(x)\geq 0$ and $v(x,0)=v_0(x)\geq 0$ produce a
unique classical and nonnegative solution $(u,v,w)$ to
\eqref{problem_abstract}, which is defined in $\Omega \times (0,T_{max})$, we
establish that for any such $(u_0,v_0)$, the life span $\TM=\infty$ and $u, v$
and $w$ are uniformly bounded in $\Omega\times (0,\infty)$, (i) for $l=1$,
$n\in \{1,2\}$, $\alpha\in (0,\frac{1}{2}+\frac{1}{n})\cap (0,1)$ and any
$\xi>0$, (ii) for $l=1$, $n\geq 3$, $\alpha\in (0,\frac{1}{2}+\frac{1}{n})$ and
$\xi$ larger than a quantity depending on $\chi \lVert v_0
\rVert_{L^\infty(\Omega)}$, (iii) for $l>1$ any $\xi>0$, and in any dimensional
settings. Finally, an indicative analysis about the effect by logistic and
repulsive actions on chemotactic phenomena is proposed by comparing the results
herein derived for the linear production case with those in
\cite{LankeitWangConsumptLogistic}.