Blow-up phenomena for a chemotaxis system with flux limitation

Abstract

In this paper we consider nonnegative solutions of the following parabolic-elliptic cross-diffusion system{ut=Δu−∇(uf(|∇v|2)∇v),0=Δv−μ+u,∫Ωv=0,μ:=1|Ω|∫Ωudx,u(x,0)=u0(x), in Ω×(0,∞), with Ω a ball in RN, N≥3 under homogeneous Neumann boundary conditions and f(ξ)=(1+ξ)−α, 0<α<N−22(N−1), which describes gradient-dependent limitation of cross diffusion fluxes. Under conditions on f and initial data, we prove that a solution which blows up in finite time in L∞-norm, blows up also in Lp-norm for some p>1. Moreover, a lower bound of blow-up time is derived.