Chemotactic systems in the presence of conflicts: A new functional inequality

The evolution of a chemotactic system involving a population of cells attracted to self-produced chemicals is described by the Keller–Segel system. In dimension 2, this system demonstrates a balance between the spreading effect of diffusion and the concentration due to self-attraction. As a result, there exists a critical “mass” (i.e. total cell's population) above which the solution of this system collapses in a finite time, while below this critical mass there is global existence in time. In particular, subcritical mass leads under certain additional conditions to the existence of steady states, corresponding to the solution of an elliptic Liouville equation. The existence of this critical mass is related to a functional inequality known as the Moser–Trudinger inequality.An extension of the Keller–Segel model to several cells populations was considered before in the literature. Here we review some of these results and, in particular, consider the case of conflict between two populations, that is, when population one attracts population two, while, at the same time, population two repels population one. This assumption leads to a new functional inequality which generalizes the Moser–Trudinger inequality. As an application of this inequality we derive sufficient conditions for the existence of steady states corresponding to solutions of an elliptic Liouville system.