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
Bibtex
MSC-Numbers: 35B40, 35K65, 92C17
Keywords and phrases: chemotaxis, degenerate diffusion, compact support
Download full preprint: PDF (420 kB)

Abstract:
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,σ ∈ (0,1) and ? ∈ (0,σ) 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 limt↗Tmax∥u(⋅,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).

Abstract

01.06.2023, 02:21