Mass Threshold for Infinite-time Blowup in a Chemotaxis Model with Split Population
Philippe Laurençot, Christian Stinner
Abstract
We study a chemotaxis model describing the space and time evolution in a smooth and bounded domain of $\mathbb{R}^2$ of the densities $u$ and $v$ of subpopulations of moving and static individuals of some species and the concentration $w$ of a chemoattractant. We prove that, in an appropriate functional setting, all solutions exist globally in time. Moreover, we establish the existence of a critical mass $M_c>0$ of the whole population $u+v$ such that, for $M \in (0, M_c)$, any solution is bounded, while, for almost all $M > M_c$, there exist solutions blowing up in infinite time. The building block of the analysis is the construction of a Liapunov functional. As far as we know, this is the first result of this kind when the mass conservation includes the two subpopulations and not only the moving one.