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

Abstract

This paper deals with the following attraction-repulsion chemotaxis system with nonlocal logistic source and sublinear productions \[ \left\{ \begin{array}{rrll} &&u_t = d_1 Δu-χ\nabla\cdot(u^k \nabla v)+ξ\nabla\cdot(u^k \nabla w)+ μu^m \left(1-\int_Ωu(x,t){\rm d}x\right),\qquad &x\inΩ,\, t>0,\\ &&v_t = d_2 Δv-αv+f(u), &x\inΩ,\, t>0,\\ &&w_t = d_3 Δw-βw+f(u), &x\inΩ,\, t>0,\\ &&\frac{\partial u}{\partialν} = \frac{\partial v}{\partialν} = \frac{\partial w}{\partialν} = 0, &x\in\partialΩ,\, t>0,\\ &&u(x,0) = u_0, \quad v(x,0)=v_0, \quad w(x,0)=w_0,&x\inΩ, \end{array} \right. \] in an open, bounded domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$, $n\ge 2$ with smooth boundary $\partial \hat{I}\text{c◯}$. Assume the parameters $d_1$, $d_2$, $d_3$, $\ddot{I}‡$, $\hat{I}¾$, $\hat{I}\pm$, ${\hat{I}}^2$ and $\hat{I}¼$ are positive constants, initial data $(u_0,v_0,w_0)$ are nonnegative and the function $f(u)\le K{u}^l\in C^1([0,\infty ))$ 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.