A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blow-up
A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blow-up
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DOI：
10.1080/03605302.2016.1277237
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发表时间：
2016-05
影响因子：
1.9
通讯作者：
N. Bellomo;M. Winkler
中科院分区：
文献类型：
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作者：
N. Bellomo;M. Winkler
摘要本文旨在为一类新的趋化性模型的定性理论提供第一步，这类趋化性模型来自著名的Keller-Segel系统，其主要新颖之处在于扩散是非线性的，具有通量的特征。更精确地说，作为这类问题的典型代表，我们研究了抛物-椭圆方程组在初始条件和无通量边界条件下的径向对称解，其中χ>0，且.主要结果是证明了存在唯一的经典解，该解在时间上可扩展到极大的Tmax∈（0，∞]，它具有以下性质：这个证明主要是基于比较方法，首先将空间梯度ur的逐点上下界与u的L∞界和u的上界联系起来;第二，涉及非局部非线性的另一个比较论证提供了z+在u的界限方面的适当控制，|ur|对后者有适度的轻度依赖。作为（1）的结果，通过适当的先验估计，进一步证明了上述解是全局的和有界的，当x>1和mc：= ∞，如果x ≤1。这些条件本质上是最优的，这将在即将发表的一篇论文中得到证明，其中，（Ω）将被用来导出关于u在L∞（Ω）中的范数在有限时间内爆破的解的出现的补充结果。
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 under the initial condition and no-flux boundary conditions in balls Ω⊂ℝn, where χ>0 and . The main results assert the existence of a unique classical solution, extensible in time up to a maximal Tmax∈(0,∞] which has the property that The proof of this is mainly based on comparison methods, which first relate pointwise lower and upper bounds for the spatial gradient ur to L∞ bounds for u and to upper bounds for ; second, another comparison argument involving nonlocal nonlinearities provides an appropriate control of z+ in terms of bounds for u and |ur|, 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 with if χ>1 and mc: = ∞ if χ≤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∞(Ω).