Nonlinear stability of strong traveling waves for the singular Keller–Segel system with large perturbations

Abstract

This paper is concerned with the nonlinear stability of traveling wave solutions for a conserved system of parabolic equations derived from a singular chemotaxis model describing the initiation of tumor angiogenesis. When the initial datum is a continuous small perturbation with zero integral from the spatially shifted traveling wave, the asymptotic stability of the large-amplitude (strong) traveling waves has been established in a series of works [29,34,35] by the second author with his collaborators. In this paper, we shall show that similar stability results indeed hold true for large and discontinuous initial data (i.e. the initial perturbation from the traveling wave could be discontinuous and has large oscillations) such as Riemann data with large jumps. To the best of our knowledge, this paper provides a first result on the asymptotic stability of large-amplitude traveling waves with large initial perturbation for a system of conservation laws, although similar results have been available for the scalar equations (cf. [8,42]). We also extend existing results to the initial data with lower regularity.