Taxis-driven persistent localization in a degenerate Keller-Segel system. Issue 12 (2nd December 2022) - Stevens, Angela; Winkler, Michael
Abstract: The degenerate Keller-Segel type system { ∂ t u = ∇ · ( u m − 1 ∇ u ) − ∇ · ( u ∇ v ), x ∈ Ω,   t > 0, 0 = Δ v − μ + u,     ∫ Ω v = 0,   μ = 1 | Ω | ∫ Ω u,     x ∈ Ω,   t > 0, is considered in balls Ω = B R ( 0 ) ⊂ ℝ n 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 θ ∈ ( 0, σ ) one can find R ⋆ = R ⋆ ( n, m, μ, σ, θ ) > 0 such that if R ≥ R ⋆ and u 0 ∈ L ∞ ( Ω ) is nonnegative and radially symmetric with 1 | Ω | ∫ Ω u 0 = μ and 1 | B r ( 0 ) | ∫ B r ( 0 ) u 0 ≥ μ θ n     for   all   r ∈ ( 0, θ R ), then a corresponding zero-flux type initial-boundary value problem admits a radial weak solution ( u, v ), extensible up to a maximal time T max ∈ ( 0, ∞ ] and satisfying lim ⁡ t ↗ T max ‖ u ( ·, t ) ‖ L ∞ ( Ω ) = ∞ if T max < ∞, which has the additional property that supp   u ( ·, t ) ⊂ B ¯ σ R ( 0 )     for   all   t ∈ ( 0, T max ) . In particular, this conclusion is seen to be valid whenever u 0 is radially nonincreasing with supp   u 0 ⊂ B ¯ θ R ( 0 ) .