Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model

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Abstract

We consider the classical parabolic-parabolic Keller-Segel system. {ut=Δu-∇{dot operator}(u∇v),x∈Ω,t>0,vt=Δv-v+u,x ∈Ω,t>0, under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂R{double-struck}n. It is proved that in space dimension n≥3, for each q>n/2 and p>n one can find 0>0 such that if the initial data (u0,v0) satisfy {norm of matrix}u0{norm of matrix}Lq(Ω)< and {norm of matrix}∇v0{norm of matrix}Lp(Ω)< then the solution is global in time and bounded and asymptotically behaves like the solution of a discoupled system of linear parabolic equations. In particular, (u,v) approaches the steady state (m,m) as t→∞, where m is the total mass m:=∫Ωu0 of the population. Moreover, we shall show that if Ω is a ball then for arbitrary prescribed m>0 there exist unbounded solutions emanating from initial data (u0,v0) having total mass ∫Ωu0=m. © 2010 Elsevier Inc.

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Winkler, M. (2010). Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model. Journal of Differential Equations, 248(12), 2889–2905. https://doi.org/10.1016/j.jde.2010.02.008

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