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

TL;DR

This paper establishes the existence and long-term behavior of weak solutions for a singular PDE-ODE chemotaxis system modeling tumor angiogenesis, accommodating discontinuous initial data using innovative energy estimates.

Contribution

It introduces the use of effective viscous flux to prove asymptotic stability for chemotaxis systems with discontinuous initial conditions, extending prior results.

Findings

Solutions converge to equilibrium as time tends to infinity.

Effective viscous flux enables energy estimates for low-regularity data.

Asymptotic stability proven for discontinuous initial data.

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 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 datum 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 so-called "effective viscous flux", which enables us to obtain the desired energy estimates and regularity. The technique of "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.