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首页> 外文期刊>Discrete and continuous dynamical systems▼hSeries S >GLOBAL EXISTENCE IN THE 1D QUASILINEAR PARABOLIC-ELLIPTIC CHEMOTAXIS SYSTEM WITH CRITICAL NONLINEARITY
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GLOBAL EXISTENCE IN THE 1D QUASILINEAR PARABOLIC-ELLIPTIC CHEMOTAXIS SYSTEM WITH CRITICAL NONLINEARITY

机译：具有临界非线性的1D拟线性抛物线 - 椭圆趋化系统的全局存在

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摘要

The paper should be viewed as complement of an earlier result in [10]. In the paper just mentioned it is shown that 1d case of a quasilinear parabolic-elliptic Keller-Segel system is very special. Namely, unlike in higher dimensions, there is no critical nonlinearity. Indeed, for the nonlinear diffusion of the form 1/u all the solutions, independently on the magnitude of initial mass, stay bounded. However, the argument presented in [10] deals with the Jaeger-Luckhaus type system. And is very sensitive to this restriction. Namely,the change of variables introduced in [10], being a main step of the method,works only for the Jaeger-Luckhaus modification. It does not seem to be applicable in the usual version of the parabolic-elliptic Keller-Segel system. The present paper fulfils this gap and deals with the case of the usual parabolicelliptic version. To handle it we establish a new Lyapunov-like functional (it is related to what was done in [10]), which leads to global existence of the initial-boundary value problem for any initial mass.
机译：本文应被视为[10]的早期结果的补充。在本文中，刚提到了它表明，Quasilinear抛物线 - 椭圆形Keller-Segel系统的1D案例非常特别。即，与较高尺寸不同，没有关键的非线性。实际上，对于形式1 / u的非线性扩散，所有溶液的初始质量的大小独立地保持界定。但是，[10]中提出的论点处理Jaeger-Luckhaus型系统。并且对这种限制非常敏感。即，[10]中引入的变量的变化是该方法的主要步骤，仅适用于Jaeger-Luckhaus修改。它似乎并不适用于抛物线椭圆凯勒-Segel系统的通常版本。本文履行了这种差距并处理了通常的抛物面版本的情况。为了处理它，我们建立了一个新的Lyapunov样功能（它与[10]中的内容有关，这导致全局存在于任何初始质量的初始边界值问题。

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