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 $\left(\begin{array}{cc}{u}_t=\nabla \cdot \left(\frac{u\nabla u}{\sqrt{u^2+{|\nabla u|}^2}}\right)-\chi k_f\nabla \cdot \left(\frac{u\nabla v}{{(1+{|\nabla v|}^2)}^{\alpha }}\right),& x\in \Omega ,t>0,\\ 0=\Delta v-\mu +u,& x\in \Omega ,t>0\end{array}\right)$under no flux boundary conditions in a ball $B=\Omega \subset R^N$ and initial condition $u(x,0)=u_0\left(x\right)>0,\chi >0,\alpha >0,k_f>0$ and $\mu =\frac{1}{\left|\Omega \right|}{\int }_{\Omega }{u}_{0}dx.$ Under suitable conditions on $\alpha$ and $u_0$ it is shown that the solution blows up in $L^{\infty }$-norm at a finite time $T_{max}$ and for some $p>1$ it blows up also in $L^p$-norm. The proofs are mainly based on an helpful change of variables, on comparison arguments and some suitable estimates.