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

TL;DR

This paper proves the global well-posedness of large perturbations around traveling waves in a hyperbolic-parabolic chemotaxis system, extending stability results to arbitrarily large initial data without mean-zero restrictions.

Contribution

It develops a new energy estimate and uses a relative entropy approach to establish global existence for large perturbations, regardless of mean-zero conditions.

Findings

Global existence of solutions for large perturbations in $H^1$

Stability of traveling waves with small shock strength

Use of a priori contraction estimates and relative entropy methods

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 stabilities have been considered in the class of certain mean-zero small perturbations. We show the global existence of the 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 wave 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, 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. The boundedness of the shift implies a priori bound of the relative entropy functional without a 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.