Global existence and boundedness in an attraction-repulsion chemotaxis system with nonlocal logistic source and sublinear productions

TL;DR

This paper proves the global existence and boundedness of solutions for a complex chemotaxis system involving attraction, repulsion, nonlocal logistic growth, and sublinear production terms in bounded domains.

Contribution

It establishes conditions on parameters ensuring the existence of unique, globally bounded classical solutions for a chemotaxis model with nonlocal and nonlinear effects.

Findings

Global bounded classical solutions exist under certain parameter conditions.

The model incorporates nonlocal logistic source and sublinear production terms.

Conditions on parameters k, l, m are derived for solution boundedness.

Abstract

This paper deals with the following attraction-repulsion chemotaxis system with nonlocal logistic source and sublinear productions \[ \left\{ \begin{array}{rrll} &&u_t = d_1 \Delta u-\chi \nabla\cdot(u^k \nabla v)+\xi \nabla\cdot(u^k \nabla w)+ \mu u^m \left(1-\int_\Omega u(x,t){\rm d}x\right),\qquad &x\in\Omega,\, t>0,\\ &&v_t = d_2 \Delta v-\alpha v+f(u), &x\in\Omega,\, t>0,\\ &&w_t = d_3 \Delta w-\beta w+f(u), &x\in\Omega,\, t>0,\\ &&\frac{\partial u}{\partial\nu} = \frac{\partial v}{\partial\nu} = \frac{\partial w}{\partial\nu} = 0, &x\in\partial\Omega,\, t>0,\\ &&u(x,0) = u_0, \quad v(x,0)=v_0, \quad w(x,0)=w_0,&x\in\Omega, \end{array} \right. \] in an open, bounded domain $\Omega \subset \mathbb{R}^n$, $n\geq 2$ with smooth boundary $\partial\Omega$. Assume the parameters $d_1$, $d_2$, $d_3$, $\chi$, $\xi$, $\alpha$, $\beta$ and $\mu$ are positive constants, initial data $(u_0, v_0, w_0)$ are nonnegative and the function $f(u)\leq K u^l\in C^1([0, \infty))$ for some $K, l>0$. Under appropriate conditions on the parameter $k$, $l$ and $m$ we show that the above problem admits a unique globally bounded classical solution.