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MASS THRESHOLD FOR INFINITE-TIME BLOWUP IN A CHEMOTAXIS MODEL WITH SPLITTED POPULATION

Abstract : We study the chemotaxis model ∂ t u = div(∇u − u∇w) + θv − u in (0, ∞) × Ω, ∂ t v = u − θv in (0, ∞) × Ω, ∂ t w = D∆w − αw + v in (0, ∞) × Ω, with no-flux boundary conditions in a bounded and smooth domain Ω ⊂ 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 ∈ (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.
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https://hal.archives-ouvertes.fr/hal-02943355
Contributor : Philippe Laurençot <>
Submitted on : Saturday, September 19, 2020 - 9:59:20 AM
Last modification on : Tuesday, September 22, 2020 - 3:29:12 AM

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  • HAL Id : hal-02943355, version 1
  • ARXIV : 2009.09655

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Philippe Laurençot, Christian Stinner. MASS THRESHOLD FOR INFINITE-TIME BLOWUP IN A CHEMOTAXIS MODEL WITH SPLITTED POPULATION. 2020. ⟨hal-02943355⟩

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