A free boundary problem for a class of parabolic-elliptic type chemotaxis model

TL;DR

This paper investigates a free boundary problem for a chemotaxis model of parabolic-elliptic type, establishing existence of solutions, deriving explicit free boundary formulas, and demonstrating chemotactic collapse in high-dimensional symmetric domains.

Contribution

It introduces a novel approach to solving free boundary problems in chemotaxis models using contraction mapping and operator semigroup methods, with explicit boundary formulas and collapse analysis.

Findings

Existence of solutions in high-dimensional symmetric domains.

Explicit formula for the free boundary.

Demonstration of chemotactic collapse.

Abstract

In this paper, we study a free boundary problem for a class of parabolic-elliptic type chemotaxis model in high dimensional symmetry domain \Omega. By using the contraction mapping principle and operator semigroup approach, we establish the existence of the solution for such kind of chemotaxis system in the domain \Omega with free boundary condition. Besides, we get the explicit formula for the free boundary and show the chemotactic collapse for the solution of the system.