Nonlinear stability of diffusive contact wave for a chemotaxis model
Nonlinear stability of diffusive contact wave for a chemotaxis model
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DOI：
10.1016/j.jde.2021.11.008
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发表时间：
2022-01
影响因子：
2.4
通讯作者：
Yanni Zeng
中科院分区：
文献类型：
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作者：
Yanni Zeng
考虑一个2× 2双曲-抛物平衡律组。我们的系统是Keller-Segel型趋化性模型的逆Hopf-Cole变换下的转换形式，该模型具有逻辑增长、对数敏感性、非扩散化学信号和密度依赖的生产/消耗速率。研究了当柯西数据在扩散接触波附近时的柯西问题。当x→±∞时，接触波连接两个不同的终态，反映了对数奇异性在原始趋化性模型中发挥内在作用的情况。我们建立了解的整体存在性，并研究了解的时间渐近性态。从而得到了扩散接触波的非线性稳定性。我们的结果显示了显着的差异时，比较我们的模型与欧拉方程的阻尼。在我们的例子中，在渐近模中存在一个次波。因此，柯西问题的解收敛到扩散接触波的速度比有阻尼的欧拉方程慢。除了它自己的物理相关性，我们的模型是一个原型的一般系统的双曲抛物平衡定律。我们的结果为进一步研究一般系统的非线性稳定性提供了参考。
We consider a 2× 2 system of hyperbolic-parabolic balance laws. Our system is the converted form under inverse Hopf-Cole transformation of a Keller-Segel type chemotaxis model with logistic growth, logarithmic sensitivity, non-diffusive chemical signal and density-dependent production/consumption rate. We study Cauchy problem when the Cauchy data are near a diffusive contact wave. The contact wave connects two different end-states as x→±∞, reflecting the situation when the logarithmic singularity plays an intrinsic role in the original chemotaxis model. We establish global existence of solution and study time asymptotic behavior of the solution. Consequently, we obtain nonlinear stability of the diffusive contact wave. Our result shows a significant difference when comparing our model to Euler equations with damping. In our case, there exists a secondary wave in the asymptotic ansatz. Therefore, the solution to Cauchy problem converges to the diffusive contact wave slower than in the case of Euler equations with damping. Besides its own physical relevance, our model is a prototype of a general system of hyperbolic-parabolic balance laws. Our results shed light on the future study of nonlinear stability of elementary waves for a general system.