Blow-up phenomena for a chemotaxis system with flux limitation

In this paper we consider nonnegative solutions of the following parabolic-elliptic cross-diffusion system \begin{equation*} \left\{ \begin{array}{l} \begin{aligned} &u_t = \Delta u - \nabla(u f(|\nabla v|^2 )\nabla v), \\[6pt] &0= \Delta v -\mu + u , \quad \int_{\Omega}v =0, \ \ \mu := \frac 1 {|\Omega|} \int_{\Omega} u dx, \\[6pt] &u(x,0)= u_0(x), \end{aligned} \end{array} \right. \end{equation*} in $\Omega \times (0,\infty)$, with $\Omega$ a ball in $\mathbb{R}^N$, $N\geq 3$ under homogeneous Neumann boundary conditions and $f(\xi) = (1+ \xi)^{-\alpha}$, $0<\alpha < \frac{N-2}{2(N-1)}$, 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 $L^\infty$-norm, blows up also in $L^p$-norm for some $p>1$. Moreover, a lower bound of blow-up time is derived. \vskip.2truecm \noindent{\bf AMS Subject Classification }{Primary: 35B44; Secondary: 35Q92, 92C17.} \vskip.2truecm \noindent{\bf Key Words:} finite-time blow-up; chemotaxis.