Global boundedness of solutions to a two-species chemotaxis system

Abstract

In this paper, we consider the chemotaxis system of two species which are attracted by the same signal substance

$$\left\{\begin{array}{lll}u_t = \Delta u - \nabla \cdot (u \chi_1(w)\nabla w) + \mu_1 u(1 - u - a_1 v), \qquad x \in \Omega, \, t >0,\\ v_t = \Delta v - \nabla \cdot (v \chi_2(w) \nabla w) + \mu_2 v(1 - a_2u - v),\qquad x \in \Omega, \, t >0,\\ w_t = \Delta w - w + u + v, \qquad \qquad \qquad \qquad \qquad \qquad\,\,\, x \in \Omega,\, t >0 \end{array}\right.$$

under homogeneous Neumann boundary conditions in a smooth bounded domain ${\Omega \subset \mathbb{R}^n}$. We prove that if the nonnegative initial data ${(u_0, v_0) \in \big(C^0(\bar{\Omega})\big)^2}$ and ${w_0 \in W^{1, r}(\Omega)}$ for some r > n, the system possesses a unique global uniformly bounded solution under some conditions on the chemotaxis sensitivity functions χ ₁(w), χ ₂(w) and the logistic growth coefficients μ ₁, μ ₂.