Asymptotic behavior of solutions to a chemotaxis-logistic model with transitional end-states
Asymptotic behavior of solutions to a chemotaxis-logistic model with transitional end-states
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DOI：
10.1016/j.jde.2022.07.013
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发表时间：
2022
影响因子：
2.4
通讯作者：
Yanni Zeng;Kun Zhao
中科院分区：
文献类型：
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作者：
Yanni Zeng;Kun Zhao
研究了一类具有对数敏感性和密度依赖生产/消费率的Logistic增长的Keller-Segel型趋化模型的Cauchy问题.我们的柯西数据连接了化学信号的两个不同的终态，而细胞密度在远场具有其典型的承载能力。我们感兴趣的是解的时间渐近行为。我们表明，在边界线上，代表化学信号的成分收敛到一个永久的，扩散的背景波，单调连接的两个端态。另一方面，细胞成分收敛到热核的空间导数。渐近解有明确的公式，是共同的所有解决方案共享相同的最终状态。得到了最优L2和L∞收敛速度.我们首先通过Hopf-Cole逆变换将模型转换为2× 2双曲-抛物系统。然后我们应用Chapman-Enskog展开来确定渐近解。在提取渐近解之后，我们使用各种分析工具来研究剩余部分并获得最优速率。这些方法包括时间加权能量法、谱分析、绿色函数估计和迭代法。我们的结果适用于一般类的柯西数据的模型和其转换系统。特别是，我们的研究结果适用于大数据解决方案。
We study Cauchy problem of a Keller-Segel type chemotaxis model with logistic growth, logarithmic sensitivity and density-dependent production/consumption rate. Our Cauchy data connect two different end-states for the chemical signal while the cell density takes its typical carrying capacity at the far fields. We are interested in the time-asymptotic behavior of the solution. We show that in the borderline, the component representing the chemical signal converges to a permanent, diffusive background wave, which connects the two end-states monotonically. On the other hand, the cell component converges to the spatial derivative of a heat kernel. The asymptotic solution has explicit formulation and is common to all solutions sharing the same end-states. Optimal L 2 and L∞ convergence rates are obtained. We first convert the model into a 2× 2 hyperbolic-parabolic system via inverse Hopf-Cole transformation. Then we apply Chapman-Enskog expansion to identify the asymptotic solution. After extracting the asymptotic solution, we use a variety of analytic tools to study the remainder and obtain optimal rates. These include time-weighted energy method, spectral analysis, Green's function estimate and iterations. Our results apply to a general class of Cauchy data for the model and for its transformed system. In particular, our results apply to large data solutions.