Global well-posedness of large perturbations of traveling waves in a hyperbolic-parabolic system arising from a chemotaxis model

Abstract

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 H1 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 H1-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 H1-estimate thanks to a De Giorgi type lemma. Moreover, to remove possibility of vacuum appearance, we use the lemma again.