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 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 [1]. 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. (C) 2020 Elsevier Masson SAS. All rights reserved.