Mass threshold for infinite-time blowup in a chemotaxis model with splitted population

Abstract

We study the chemotaxis model $\partial$ t u = div($\nabla$u -- u$\nabla$w) + $\theta$v -- u in (0, $\infty$) x $\Omega$, $\partial$ t v = u -- $\theta$v in (0, $\infty$) x $\Omega$, $\partial$ t w = D$\Delta$w -- $\alpha$w + v in (0, $\infty$) x $\Omega$, with no-flux boundary conditions in a bounded and smooth domain $\Omega$ $\subset$ R 2 , where u and v represent the densities of subpopulations of moving and static individuals of some species, respectively, and w the concentration 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.