Behavior in time of solutions to a degenerate chemotaxis system with flux limitation

Abstract

We study a new class of Keller–Segel models, which presents a limited flux and an optimal transport of cells density according to chemical signal density. As a prototype of this class we study radially symmetric solutions to the parabolic–elliptic system ut=∇⋅(u∇uu2+|∇u|2)−χkf∇⋅(u∇v(1+|∇v|2)α),x∈Ω,t>0,0=Δv−μ+u,x∈Ω,t>0under no flux boundary conditions in a ball B=Ω⊂RN and initial condition u(x,0)=u0(x)>0,χ>0,α>0,kf>0 and μ=1|Ω|∫Ωu0dx. Under suitable conditions on α and u0 it is shown that the solution blows up in L∞-norm at a finite time Tmax and for some p>1 it blows up also in Lp-norm. The proofs are mainly based on an helpful change of variables, on comparison arguments and some suitable estimates.