Well-posedness of a Parabolic-hyperbolic Keller-Segel System in the Sobolev Space Framework

TL;DR

This paper proves the global well-posedness and decay properties of solutions to a 3D parabolic-hyperbolic Keller-Segel system with initial data near equilibrium, using Fourier analysis and smoothing effects.

Contribution

It establishes the well-posedness and decay behavior of solutions in Sobolev spaces for a complex Keller-Segel model, extending understanding of its long-term dynamics.

Findings

Global strong solutions exist for initial data close to equilibrium.

Chemical concentration decays exponentially under certain initial conditions.

Fourier analysis and smoothing effects are key tools in the proofs.

Abstract

We study the global strong solutions to a 3-dimensional parabolic-hyperbolic Keller-Segel model with initial data close to a stable equilibrium with perturbations belonging to $L^2(\mathbb R^3)\times H^1(\mathbb{R}^3)$. We obtain global well-posedness and decay property. Furthermore, if the mean value of initial cell density is smaller than a suitabale constant, then the chemical concentration decays exponentially to zero as $t$ goes to infinity. Proofs of the main results are based on an application of Fourier analysis method to uniform estimates for a linearized parabolic-hyperbolic system and also based on the smoothing effect of the cell density as well as the damping effect of the chemical concentration.