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A degenerate chemotaxis system with flux limitation: maximally extended solutions and absence of gradient blow-up. (English) Zbl 1430.35166
Summary: 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
\begin{aligned} u_t &= \nabla\cdot \left(\frac{u\nabla u}{\sqrt{u^2+\vert\nabla u\vert^2}}\right) - \chi\, \nabla\cdot \left(\frac{u\nabla v}{\sqrt{1+\vert\nabla v\vert^2}}\right), \\ 0 &= \Delta v -\mu + u, \end{aligned}
under the initial condition $$u\vert_{t=0} = u_0>0$$ and no-flux boundary conditions in balls $$\Omega\subset \mathbb R^n$$, where $$\chi >0$$ and $$\mu:= \frac1{\vert\Omega\vert} \int_\Omega u_0$$.
The main results assert the existence of a unique classical solution, extensible in time up to a maximal $$T_{\text{max}}\in (0,\infty]$$ which has the property that
$\text{if}\quad T_{\text{max}}<\infty\quad\text{then}\quad \limsup_{t\to T_{\text{max}}} \Vert u(\cdot, t)\Vert_{L^\infty (\Omega)} = \infty. \tag{*}$
The proof of this is mainly based on comparison methods, which first relate pointwise lower and upper bounds for the spatial gradient $$u_r$$ to $$L^\infty$$ bounds for $$u$$ and to upper bounds for $$z:= \frac{u_t}{u}$$; second, another comparison argument involving nonlocal nonlinearities provides an appropriate control of $$z_+$$ in terms of bounds for $$u$$ and $$\vert u_r\vert$$, with suitably mild dependence on the latter.
As a consequence of (*), by means of suitable a priori estimates, it is moreover shown that the above solutions are global and bounded when either $$n\ge 2$$ and $$\chi<1$$, or $$n=1$$, $$\chi>0$$ and $$m< m_c$$, with $$m_c:= \frac1{\sqrt{\chi^2-1}}$$ 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 (*) 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 $$L^\infty(\Omega)$$.

##### MSC:
 35M33 Initial-boundary value problems for mixed-type systems of PDEs 35B07 Axially symmetric solutions to PDEs 35B44 Blow-up in context of PDEs 35B45 A priori estimates in context of PDEs 35K59 Quasilinear parabolic equations 35K65 Degenerate parabolic equations 35Q92 PDEs in connection with biology, chemistry and other natural sciences 92C17 Cell movement (chemotaxis, etc.)
##### Keywords:
chemotaxis; degenerate diffusion; flux limitation
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