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

TL;DR

This work analyzes a singular chemotaxis system in dimensions two and three, proving that global solutions near equilibrium decay at optimal algebraic rates for all derivatives, with the critical high-order derivative achieving rate $(1+t)^{-(d/2 + N)/2}$. The approach combines Cole–Hopf transformation, precise spectral analysis of the linearized semigroup, Fourier splitting, and energy methods to derive both upper and lower bounds, thereby matching heat-kernel decay. The results are then lifted back to the original variables, yielding sharp $L^\infty$ decay for the density and exponential decay for the chemical concentration. Overall, the paper provides a complete, optimal decay theory for the nonlinear singular chemotaxis system in multi-dimensions.

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.

Theorems & Definitions (25)

• Proposition 1.1: WXY2016
• Theorem 1.1: Upper bounds
• Remark 1.1
• Theorem 1.2: Lower bounds
• Remark 1.2
• Theorem 1.3
• Lemma 2.1: Sobolev interpolation inequality
• Lemma 2.2
• Lemma 3.1
• proof