Behavior in time of solutions of a Keller–Segel system with flux limitation and source term

Abstract

In this paper we consider radially symmetric solutions of the following parabolic–elliptic cross-diffusion system ut=Δu-∇·(uf(|∇v|2)∇v)+g(u),0=Δv-m(t)+u,∫Ωvdx=0,u(x,0)=u0(x),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} u_t = \\Delta u - \ abla \\cdot (u f(|\ abla v|^2 )\ abla v) + g(u), &{} \\\\ 0= \\Delta v -m(t)+ u, \\quad \\int _{\\Omega }v \\,dx=0, &{} \\\\ u(x,0)= u_0(x), &{} \\end{array}\\right. } \\end{aligned}$$\\end{document}in Omega times (0,infty ), with Omega a ball in {mathbb {R}}^N, Nge 3, under homogeneous Neumann boundary conditions, where g(u)= lambda u - mu u^k, lambda>0, mu >0, and k >1, f(|nabla v|^2 )= k_f(1+ |nabla v|^2)^{-alpha }, alpha >0, which describes gradient-dependent limitation of cross diffusion fluxes. The function m(t) is the time dependent spatial mean of u(x, t) i.e. m(t):= frac{1}{|Omega |} int _{Omega } u(x,t) ,dx. Under smallness conditions on alpha and k, we prove that the solution u(x, t) blows up in L^{infty }-norm at finite time T_{max} and for some p>1 it blows up also in L^p-norm. In addition a lower bound of blow-up time is derived. Finally, under largeness conditions on alpha or k, we prove that the solution is global and bounded in time.