Global weak solutions and asymptotics of a singular PDE-ODE Chemotaxis system with discontinuous data

Hongyun Peng, Zhi An Wang, Changjiang Zhu

Research output: Journal article publicationJournal articleAcademic researchpeer-review

Abstract

This paper is concerned with the well-posedness and large-time behavior of a two-dimensional PDE-ODE hybrid chemotaxis system describing the initiation of tumor angiogenesis. We first transform the system via a Cole-Hopf type transformation into a parabolic-hyperbolic system and then show that the solution of the transformed system converges to a constant equilibrium state as time tends to infinity. Finally we reverse the Cole-Hopf transformation and obtain the relevant results for the pre-transformed PDE-ODE hybrid system. In contrast to the existing related results, where continuous initial data is imposed, we are able to prove the asymptotic stability for discontinuous initial data with large oscillations. The key ingredient in our proof is the use of the so-called “effective viscous flux”, which enables us to obtain the desired energy estimates and regularity. The technique of the “effective viscous flux” turns out to be a very useful tool to study chemotaxis systems with initial data having low regularity and was rarely (if not) used in the literature for chemotaxis models.

Original languageEnglish
Pages (from-to)1-22
Number of pages22
JournalScience China Mathematics
DOIs
Publication statusE-pub ahead of print - 6 Dec 2020

Keywords

  • 35A01
  • 35B40
  • 35Q92
  • 92C17
  • asymptotic stability
  • chemotaxis
  • discontinuous initial data
  • effective viscous flux

ASJC Scopus subject areas

  • Mathematics(all)

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