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

Stevens, Angela; Winkler, Michael

Zusammenfassung

Abstract The degenerate Keller-Segel type system {∂tu0==∇⋅(um−1∇u)−∇⋅(u∇v),Δv−μ+u, ∫Ωv=0, μ=1|Ω|∫Ωu, x∈Ω, t0,x∈Ω, t0, 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 μ0,σ∈(0,1) and θ∈(0,σ) one can find R⋆=R⋆(n,m,μ,σ,θ)0 such that if R≥R⋆ and u0∈L∞(Ω) is nonnegative and radially symmetric with 1|Ω|∫Ωu0=μ and 1|Br(0)|∫Br(0)u0≥μθn for 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 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 supp u0⊂B¯¯¯θR(0).