Preprint 4/2022

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

Angela Stevens and Michael Winkler

Contact the author: Please use for correspondence this email.
Submission date: 10. Feb. 2022
Pages: 22
MSC-Numbers: 35B40, 35K65, 92C17
Keywords and phrases: chemotaxis, degenerate diffusion, compact support
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The degenerate Keller-Segel type system


{               ut& = & ∇ ⋅∫(um−1∇u )− ∇ ⋅(u∫∇v),    &x ∈ Ω, t > 0,    0& = &Δv − μ + u,    Ω v = 0, μ = 1|Ω| Ω u,    &x ∈ Ω, t > 0,

is considered in balls Ω = BR(0) Rn with n 1, R > 0 and m > 1.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(0,1) and ? (0) one can find R = R(n,m,μ,σ,?) > 0 such that if R R and u0 L(Ω) is nonnegative and radially symmetric with -1 |Ω| Ωu0 = μ and


   1  ∫          μ |Br(0)|  B(0)u0 ≥ ?n-    for all r ∈ (0,?R),          r

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


           -- supp u(⋅,t) ⊂ B σR(0)  for all t ∈ (0,Tmax).

In particular, this conclusion is seen to be valid whenever u0 is radially nonincreasing with suppu0 B?R(0).


02.02.2023, 02:21