We consider a one-dimensional system arising from a chemotaxis model in tumour angiogenesis, which is described by a Keller-Segel equation with singular sensitivity. This hyperbolic-parabolic system is known to allow viscous shocks (so-called traveling waves), and in literature, their nonlinear stability has been considered in the class of certain mean-zero small perturbations. We show the global existence of solution without assuming the mean-zero condition for any initial data as arbitrarily large perturbations around traveling waves in the Sobolev space $H^1$ while the shock strength is assumed to be small enough. The main novelty of this paper is to develop the global well-posedness of any large $H^1$-perturbations of traveling waves connecting two different end states. The discrepancy of the end states is linked to the complexity of the corresponding flux, which requires a new type of an energy estimate. To overcome this issue, we use the a priori contraction estimate of a weighted relative entropy functional up to a translation, which was proved by Choi-Kang-Kwon-Vasseur [4]. The boundedness of the shift implies a priori bound of the relative entropy functional without the shift on any time interval of existence, which produces a $H^1$-estimate thanks to a De Giorgi type lemma. Moreover, to remove possibility of vacuum appearance, we use the lemma again.

我们考虑在肿瘤血管生成中由趋化模型产生的一维系统，该系统由具有奇异敏感性的Keller-Segel方程描述。众所周知，这种双曲线-抛物线系统会产生粘性冲击（所谓的行波），在文献中，它们的非线性稳定性已在某些均值为零的小扰动类别中被考虑。我们显示了解的全局存在，而没有假设任何初始数据的均值零条件为Sobolev空间中行波周围的任意大扰动$H^{\mathrm{1个}}$而冲击强度被认为足够小。本文的主要新颖之处在于发展任何大型企业的全球定位性$H^{\mathrm{1个}}$-连接两个不同最终状态的行波摄动。最终状态的差异与相应通量的复杂性有关，这需要一种新型的能量估计。为了克服这个问题，我们使用了直至翻译的加权相对熵函数的先验收缩估计，这由Choi-Kang-Kwon-Vasseur证明[4]。位移的有界性意味着相对熵函数的先验界限，在存在的任何时间间隔上都没有位移，这会产生一个$H^{\mathrm{1个}}$-估计要感谢De Giorgi类型引理。此外，为了消除出现真空现象的可能性，我们再次使用引理。