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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.)
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