Global existence and boundedness in an attraction-repulsion chemotaxis system with nonlocal logistic source and sublinear productions
Gnanasekaran Shanmugasundaram, Nithyadevi Nagarajan
TL;DR
This work addresses the global well-posedness of a fully parabolic attraction–repulsion chemotaxis system with a nonlocal logistic source and sublinear production on a bounded domain of dimension $n\ge 2$. The authors develop a framework based on classical parabolic estimates, mass controls, and maximal regularity to derive uniform-in-time bounds, culminating in a global existence and boundedness result. The main contribution is proving the existence of a unique globally bounded classical solution under conditions $k\ge1$, $m<1+\frac{2}{n}$, and $l+k<1+\frac{2}{n}$ with $f(u)\le K u^l$, thereby extending the theory of chemotaxis models to include nonlocal logistic damping and sublinear production. This result enhances understanding of how nonlocal and sublinear mechanisms influence long-time dynamics in attraction–repulsion chemotaxis systems with fully parabolic coupling.
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 $Ω\subset \mathbb{R}^n$, $n\geq 2$ with smooth boundary $\partialΩ$. Assume the parameters $d_1$, $d_2$, $d_3$, $χ$, $ξ$, $α$, $β$ and $μ$ 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.