Asymptotic behavior of solutions to a singular chemotaxis system in multi-dimensions

TL;DR

This paper analyzes the long-term behavior of solutions to a singular chemotaxis system in two and three dimensions, showing optimal convergence rates to equilibrium using advanced mathematical techniques.

Contribution

It provides the first optimal decay rates for solutions to a singular chemotaxis system in multi-dimensions, improving previous results and establishing lower bounds matching heat equation rates.

Findings

Global solutions converge to equilibrium at optimal rates

Derived lower bounds on convergence rates for specific initial data

Used spectral analysis and Fourier splitting techniques

Abstract

In this paper, we investigate the optimal large-time behavior of the global solution to a singular chemotaxis system in the whole space $\mathbb{R}^d$ with $d=2,3$. Assuming that the initial data is sufficiently close to an equilibrium state, we first prove the $k$-th order spatial derivative of the global solution converges to its corresponding equilibrium at the optimal rate $(1+t)^{-(\frac{d}{4}+\frac{k}{2})}$, which improve upon the result in [37]. Then, for well-chosen initial data, we also establish lower bounds on the convergence rates, which match those of the heat equation. Our proof relies on a Cole-Hopf type transformation, delicate spectral analysis, the Fourier splitting technique, and energy methods.