Global attractor for a two-dimensional chemotaxis system with linear degradation and indirect signal production

Abstract

We study the asymptotic behavior of solutions to a chemotaxis system with indirect signal production presented by Deneubourg (Insectes Sociaux 24:117–130, 1977): $$\begin{aligned} \left\{ \begin{array}{ll} \;\; u_t = \varDelta \; u - \chi \nabla \cdot ( u \nabla w ) + 1 - \mu u &{}\quad \text {in } \varOmega \; \times (0,\infty ) , \qquad \\ \delta \, v_t = - v + u &{}\quad \text {in } \varOmega \; \times (0,\infty ) , \qquad \\ \tau \, w_t = \varDelta \; w - w + v &{}\quad \text {in } \varOmega \; \times (0,\infty ) . \qquad \\ \end{array}\right. \end{aligned}$$Here, $$\varOmega \; \subset {\mathbb {R}}^2$$ is a smooth bounded domain with homogeneous Neumann boundary conditions imposed on its boundary. The coefficients are all positive constants. The system models the self-organized nest construction process of social insects, specifically, termites. We first show the global-in-time existence of solutions with some smallness conditions for chemotactic intensity $$\chi $$ or the initial total mass $$\Vert u_0 \Vert _{L_1}$$ of worker insects with sufficiently large rest rate $$\mu $$ of working. We then define the dynamical system of solutions and construct the global attractor. In addition, for $$\mu / \chi $$ further large, we also construct a Lyapunov functional for the unique homogeneous equilibrium $$(1/\mu , 1/\mu , 1/\mu )$$, which indicates that the global attractor consists only of the equilibrium.