发表机构
Jiangxi Normal University; Northeast Normal University（江西师范大学; 东北师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究趋化性抛物-双曲系统中行波与稀疏波复合波的非线性稳定性，通过加权相对熵与能量结构结合，证明小扰动下全局解收敛于复合波。
AI 中文摘要
我们研究了趋化性驱动的抛物-双曲系统中由行波和稀疏波组成的复合波的非线性稳定性。我们证明，如果初始值是复合波的小型$H^1$型扰动，则该系统存在一个全局解，该解以绝对连续的平移收敛于复合波。证明结合了用于粘性激波的加权相对熵机制与稀疏波的能量结构。一个关键要素是在加权相对熵中引入稀疏波调制因子；其导数与稀疏波剖面产生的项相结合，产生稀疏波耗散。此外，两波的空间分离产生了由非精确叠加引起的时间可积相互作用误差。由于扩散仅作用于密度分量，完整的$H^1$估计通过利用系统的耦合结构来恢复双曲分量缺失的耗散而得以闭合。
英文摘要
We study the nonlinear stability of a composite wave consisting of a traveling wave and a rarefaction wave for a parabolic-hyperbolic system arising from chemotaxis. We prove that if the initial value is a small $H^1$-type perturbation of composite wave, then the system admits a global solution that converges toward the composite wave with an absolutely continuous shift. The proof combines the weighted relative-entropy mechanism for viscous shocks with the energy structure of rarefaction waves. A key ingredient is the inclusion of a rarefaction modulation factor in the weighted relative entropy; its derivatives combine with the terms generated by the rarefaction profile to produce a rarefaction dissipation. Moreover, the spatial separation of the two waves yields time-integrable interaction errors caused by the non-exact superposition. Since diffusion acts only on the density component, the full $H^1$ estimate is closed by exploiting the coupling structure of the system to recover the missing dissipation of the hyperbolic component.
