HYPERBOLIC-PARABOLIC BALANCE LAWS : ASYMPTOTIC BEHAVIOR AND A CHEMOTAXIS MODEL
HYPERBOLIC-PARABOLIC BALANCE LAWS : ASYMPTOTIC BEHAVIOR AND A CHEMOTAXIS MODEL
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双曲-抛物线平衡定律：渐近行为和趋化性模型
DOI：
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发表时间：
2018
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通讯作者：
Yanni Zeng
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作者：
Yanni Zeng
本文考虑一般双曲抛物平衡律组的柯西问题。当Cauchy数据是一个常数平衡态的小扰动时，作者提出了一组结构性条件，这些条件导致解的整体存在性、Lp（p ≥ 2）衰减率和渐近性态.在调查这些最近的结果，我们专注于一个KellerSegel型趋化性模型与逻辑增长项。我们以该模型为例，说明了一维空间和多维空间在时间渐近行为上的区别。对于一维空间中的模型，我们构造了一个显式的时间渐近解，并得到了更快的衰减率的渐近解的解的速度相比，常数平衡状态的解决方案。这表明，在一维空间中，如果系统是非线性的，时间渐近解需要包含Burgers方程的解。相比之下，对于多维空间，渐近解可以是线性系统的解。AMS科目分类：35B40、35M31、35Q35、35Q92
We consider Cauchy problem of a general system of hyperbolicparabolic balance laws. The author has proposed a set of structural conditions, which lead to global existence, Lp (p ≥ 2) decay rates and asymptotic behavior of solution when the Cauchy data are small perturbations of a constant equilibrium state. After surveying these recent results, we focus on a KellerSegel type chemotaxis model with a logistic growth term. We use this model as an example to illustrate the difference between one space dimension and multi space dimensions in time asymptotic behavior. For the model in one space dimension we construct a time asymptotic solution in explicit formulation, and obtain faster decay rates of the solution to the asymptotic solution, in comparison with the rates of the solution to the constant equilibrium state. This shows that in one space dimension, a time asymptotic solution needs to include solutions of Burgers equation if the system is nonlinear in nature. By contrast, for multi space dimensions asymptotic solutions can be solutions to linear systems. AMS Subject Classification: 35B40, 35M31, 35Q35, 35Q92