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MiS Preprint
4/2022

Taxis-driven persistent localization in a degenerate Keller-Segel system

Angela Stevens and Michael Winkler

Abstract

The degenerate Keller-Segel type system {ut=(um1u)(uv),xΩ, t>0,0=Δvμ+u,Ωv=0,μ=1|Ω|Ωu,xΩ, t>0, is considered in balls Ω=BR(0)Rn with n1, R>0 and m>1.\abs Our main results reveal that throughout the entire degeneracy range m(1,), the interplay between degenerate diffusion and cross-diffusive attraction herein can enforce persistent localization of solutions inside a compact subset of Ω, no matter whether solutions remain bounded or blow up. More precisely, it is shown that for arbitrary μ>0,σ(0,1) and θ(0,σ) one can find R=R(n,m,μ,σ,θ)>0 such that if RR and u0L(Ω) is nonnegative and radially symmetric with 1|Ω|Ωu0=μ and 1|Br(0)|Br(0)u0μθnfor all r(0,θR), then a corresponding zero-flux type initial-boundary value problem admits a radial weak solution (u,v), extensible up to a maximal time Tmax(0,] and satisfying limtTmaxu(,t)L(Ω)= if Tmax<, which has the additional property that suppu(,t)BσR(0)for all t(0,Tmax). In particular, this conclusion is seen to be valid whenever u0 is radially nonincreasing with suppu0BθR(0).

Received:
10.02.22
Published:
13.02.22
MSC Codes:
35B40, 35K65, 92C17
Keywords:
chemotaxis, degenerate diffusion, compact support

Related publications

inJournal
2022 Repository Open Access
Angela Stevens and Michael Winkler

Taxis-driven persistent localization in a degenerate Keller-Segel system

In: Communications in partial differential equations, 47 (2022) 12, pp. 2341-2362