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 $$\mbox{ if} \quad T_{ max}<\infty \quad \mbox{ then} \quad\limsup_{ t\nearrow T_{ max}} \|u(\cdot,t)\|_{ L^\infty(\Omega)}=\infty. \qquad \qquad (\star)$$ 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 $$n\ge 2 \ \mbox{ and } \chi<1, \qquad \mbox{ or} \qquad n=1, \ \chi>0 \ \mbox{ and } m<m_c, $$ 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 $L^\infty(\Omega)$.
Categories: math.AP