Finite-time blow-up in a quasilinear system of chemotaxis
Finite-time blow-up in a quasilinear system of chemotaxis
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DOI：
10.1088/0951-7715/21/5/009
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发表时间：
2008-05
期刊：
影响因子：
1.7
通讯作者：
Tomasz Cieślak;M. Winkler
中科院分区：
文献类型：
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作者：
Tomasz Cieślak;M. Winkler
本文考虑一类Keller-Segel型椭圆-抛物方程组，其中含有非线性扩散项.我们发现一个临界指数的非线性扩散，测量扩散的强度在点的高（人口）密度，区分有限时间爆破和全球的时间存在的一致有界的解决方案。这个临界指数取决于空间维数n = 1，除了物理上相关的情况n = 2和n = 3，在一维设置中得到的结果也可能是数学上的兴趣：在这里，即，有限时间爆炸的解决方案发生，尽管与系统相关的李雅普诺夫泛函是有界的。此外，这个一维情形是一个例子，说明了即使假定漂移项的梯度有界，也不能期望非一致抛物型漂移扩散方程解的L∞估计。
We consider an elliptic–parabolic system of the Keller–Segel type which involves nonlinear diffusion. We find a critical exponent of the nonlinearity in the diffusion, measuring the strength of diffusion at points of high (population) densities, which distinguishes between finite-time blow-up and global-in-time existence of uniformly bounded solutions. This critical exponent depends on the space dimension n ⩾ 1, where apart from the physically relevant cases n = 2 and n = 3 also the result obtained in the one-dimensional setting might be of mathematical interest: here, namely, finite-time explosion of solutions occurs although the Lyapunov functional associated with the system is bounded from below. Additionally this one-dimensional case is an example to show that L∞ estimates of solutions to non-uniformly parabolic drift–diffusion equations cannot be expected even when boundedness of the gradient of the drift term is presupposed.