Taxis-driven persistent localization in a degenerate Keller-Segel system
Angela Stevens and Michael Winkler
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Submission date: 10. Feb. 2022
MSC-Numbers: 35B40, 35K65, 92C17
Keywords and phrases: chemotaxis, degenerate diffusion, compact support
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The degenerate Keller-Segel type system
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 ∫ Ωu0 = μ and
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
In particular, this conclusion is seen to be valid whenever u0 is radially nonincreasing with suppu0 ⊂B?R(0).