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

Abstract

Abstract The degenerate Keller-Segel type system is considered in balls with R > 0 and m > 1. Our main results reveal that throughout the entire degeneracy range 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 and one can find such that if and is nonnegative and radially symmetric with and then a corresponding zero-flux type initial-boundary value problem admits a radial weak solution (u, v), extensible up to a maximal time and satisfying if which has the additional property that In particular, this conclusion is seen to be valid whenever u 0 is radially nonincreasing with