Global attractor for a two-dimensional chemotaxis system with linear degradation and indirect signal production (Q2300953)

The following system of chemotaxis-type with the indirect signal production \[ \begin{aligned} u_t&=\Delta u-\chi\nabla\cdot(u\nabla w) +1 - \mu u,\\ \delta v_t&=-v+u,\\ \tau w_t&=\Delta w-w+v, \end{aligned} \] proposed by \textit{J. L. Deneubourg} in [``Application de l'ordre par fluctuations à la description de certaines étapes de la construction du nid chez les termites'', Insectes Sociaux 24, 117--130 (1977)], is used to model the self-organized nest construction by termites. A difficulty in determination of the asymptotics of solutions is related to the \textit{partly dissipative} character of that system which makes compactness of trajectories arguments delicate. The authors study this system in bounded planar domains, and determine the long-time behavior of solutions described by the global attractor in some cases (\(\chi\) small, \(\mu\) large).