A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blowup
Abstract
This paper aims at providing a first step toward a qualitative theory for a new class of chemotaxis models derived from the celebrated KellerSegel 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 parabolicelliptic system (see the text). Under the initial condition $u_{t=0}=u_0>0$ and noflux 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 $m_c:=\frac{1}{\sqrt{\chi^21}}$ 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 $L^\infty(\Omega)$.
 Publication:

arXiv eprints
 Pub Date:
 May 2016
 arXiv:
 arXiv:1605.01924
 Bibcode:
 2016arXiv160501924B
 Keywords:

 Mathematics  Analysis of PDEs