Unique continuation inequalities for the parabolic-elliptic chemotaxis system

TL;DR

This paper establishes quantitative unique continuation inequalities for a simplified chemotaxis system, demonstrating that vanishing of one component in a subset implies the entire solution is zero, using frequency function and localization techniques.

Contribution

It introduces new unique continuation inequalities for a semi-linear parabolic-elliptic chemotaxis system, employing novel a priori estimates and analytical methods.

Findings

Proves that vanishing of one component implies the entire solution is zero.

Develops two quantitative unique continuation inequalities for the system.

Uses frequency function and localization techniques to achieve results.

Abstract

This paper studies the quantitative unique continuation for a semi-linear parabolic-elliptic coupled system on a bounded domain. This system is a simplified version of the chemotaxis model introduced by Keller and Segel. With the aid of priori L^infty-estimates (for solutions of the system) built up in this paper, we treat the semi-linear parabolic equation in the system as a linear parabolic equation, and then use the frequency function method and the localization technique to build up two unique continuation inequalities for the system. As a consequence of the above-mentioned two inequalities, we have the following qualitative unique continuation property: if one component of a solution vanishes in a nonempty open subset at some time T>0, then the solution is identically zero.