On a Parabolic-Hyperbolic Chemotaxis System with Discontinuous Data: Well-posedness, Stability and Regularity

TL;DR

This paper investigates the well-posedness, stability, and regularity of a parabolic-hyperbolic chemotaxis system with discontinuous initial data, demonstrating global solutions and their asymptotic behavior, supported by numerical simulations.

Contribution

It extends previous results by allowing large, discontinuous initial data and establishes existence, convergence, and regularity properties of solutions for the chemotaxis PDE system.

Findings

Global weak solutions exist for large discontinuous data.

Solutions converge to constant states or shock waves depending on initial asymptotic states.

Numerical simulations support analytical results and suggest minimal regularity for classical solutions.

Abstract

The global dynamics and regularity of parabolic-hyperbolic systems is an interesting topic in PDEs due to the coupling of competing dissipation and hyperbolic effects. This paper is concerned with the Cauchy problem of a parabolic-hyperbolic system derived from a chemotaxis model describing the dynamics of the initiation of tumor angiogenesis. It is shown that, as time tends to infinity, the Cauchy problem with large-amplitude discontinuous data admit global weak solutions which converge to a constant state (resp. a viscous shock wave) if the asymptotic states of initial values at far field are equal (resp. unequal). Our results improve the previous results where initial value was required to be continuous and have small amplitude. Numerical simulations are performed to verify our analytical results, illustrate the possible regularity of solutions and speculate the minimal regularity of initial data required to obtain the smooth (classical) solutions of the concerned parabolic-hyperbolic system.