Convergence to a diffusive contact wave for solutions to a system of hyperbolic balance laws
Convergence to a diffusive contact wave for solutions to a system of hyperbolic balance laws
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DOI：
10.1142/s0219891623500078
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发表时间：
2023-03
影响因子：
0.7
通讯作者：
Yanni Zeng
中科院分区：
文献类型：
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作者：
Yanni Zeng
我们考虑一个[公式：见文本]双曲平衡律系统，它是Keller-Segel型趋化模型在逆Hopf-Cole变换下的转换形式。当柯西数据连接两个不同的终态时，我们研究柯西问题[公式：见文]。背景波是约简系统的扩散接触波。建立了解的整体存在性，并研究了解的时间渐近性。在胞群初始接近稳定平衡值的特殊情况下，我们得到了小假设下扩散接触波的非线性稳定性。在一般情况下，种群最初不接近其稳定的平衡值，至少在一个远场，我们使用一个修正函数在时间渐近的方差分析，并表明群体接近逻辑稳定的平衡值。与有阻尼的欧拉方程相比，我们的结果有两个显著的不同。第一个是在时间渐近解中存在二次波。这意味着我们的解收敛到扩散接触波的速度比有阻尼的欧拉方程慢。第二个是修正函数逻辑增长而不是指数衰减。
We consider a [Formula: see text] system of hyperbolic balance laws that is the converted form under inverse Hopf–Cole transformation of a Keller–Segel type chemotaxis model. We study Cauchy problem when Cauchy data connect two different end-states as [Formula: see text]. The background wave is a diffusive contact wave of the reduced system. We establish global existence of solution and study the time asymptotic behavior. In the special case where the cellular population initially approaches its stable equilibrium value as [Formula: see text], we obtain nonlinear stability of the diffusive contact wave under smallness assumption. In the general case where the population initially does not approach to its stable equilibrium value at least at one of the far fields, we use a correction function in the time asymptotic ansatz, and show that the population approaches logistically to its stable equilibrium value. Our result shows two significant differences when comparing to Euler equations with damping. The first one is the existence of a secondary wave in the time asymptotic ansatz. This implies that our solutions converge to the diffusive contact wave slower than those of Euler equations with damping. The second one is that the correction function logistically grows rather than exponentially decays.