We study the chemotaxis model $partial$ t u = div($ abla$u -- u$ abla$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.
تحميل البحث