A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blow-up

Abstract

This paper aims at providing a first step toward a qualitative theory for a new class of chemotaxis models derived from the celebrated Keller-Segel system, with the main novelty being that diffusion is nonlinear with flux delimiter features. More precisely, as a prototypical representative of this class we study radially symmetric solutions of the parabolic-elliptic system (see the text). Under the initial condition $u{|}_{t=0}=u_0>0$ and no-flux boundary conditions in balls $\Omega \subset {\mathbb{R}}^n$, where $\chi >0$ and $\mu :=\frac{1}{|\Omega |}{\int }_{\Omega }{u}_0$.\abs The main results assert the existence of a unique classical solution, extensible in time up to a maximal $T_{max}\in (0,\infty ]$ which has the property that $$\mbox{if} \quad T_{max}<\infty \quad \mbox{then} \quad\limsup_{t\nearrow T_{max}} \|u(\cdot,t)\|_{L^\infty(Ω)}=\infty. \qquad \qquad (\star)$$ The proof therefore is mainly based on comparison methods, which firstly relate pointwise lower and upper bounds for the spatial gradient $u_r$ to $L^{\infty }$ bounds for $u$ and to {\em upper bounds} for $z:=\frac{u_t}{u}$; secondly, another comparison argument involving nonlocal nonlinearities provides an appropriate control of $z_{+}$ in terms of bounds for $u$ and $|u_r|$, with suitably mild dependence on the latter. As a consequence of ($\star$) by means of suitable a priori estimates it is moreover shown that the above solutions are global and bounded when either $$n\ge 2 \ \mbox{ and } χ<1, \qquad \mbox{or} \qquad n=1, \ χ>0 \ \mbox{ and } m<m_c, $$ with $m_c:=\frac{1}{\sqrt{{\chi }^2-1}}$ if $\chi >1$ and $m_c:=\infty$ if $\chi \le 1$. That these conditions are essentially optimal will be shown in a forthcoming paper in which ($\star$) will be used to derive complementary results on the occurrence of solutions blowing up in finite time with respect to the norm of $u$ in $L^{\infty }(\Omega )$.