Exponential grow-up rates in a quasilinear Keller–Segel system

Article type: Research Article

Authors: Winkler, Michael^{; *}

Affiliations: Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany

Correspondence: [*] Corresponding author. E-mail: michael.winkler@math.uni-paderborn.de.

Abstract: The chemotaxis system (⋆)ut=∇·(D(u)∇u)−∇·(uS(u)∇v),0=Δv−μ+u,μ=1|Ω|∫Ωu, is considered in a ball Ω=BR(0)⊂Rn. It is shown that if S∈C2([0,∞)) suitably generalizes the prototype given by S(ξ)=χξ+1,ξ⩾0, with some χ>0, and if diffusion is suitably weak in the sense that 0<D∈C2((0,∞)) is such that there exist KD>0 and m∈(−∞,1−2n) fulfilling D(ξ)⩽KDξm−1for all ξ>0, then for appropriate choices of sufficiently concentrated initial data, an associated no-flux initial-boundary value problem admits a global classical solution (u,v) which blows up in infinite time and satisfies 1Ceχt⩽‖u(·,t)‖L∞(Ω)⩽Ceχtfor all t>0. A major part of the proof is based on a comparison argument involving explicitly constructed subsolutions to a scalar parabolic problem satisfied by mass accumulation functions corresponding to solutions of (⋆).

Keywords: Chemotaxis, singularity formation, grow-up rate

DOI: 10.3233/ASY-221765

Journal: Asymptotic Analysis, vol. 131, no. 1, pp. 33-57, 2023