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

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.