Well-Posedness of the Keller–Segel System in Fourier–Besov–Morrey Spaces

Abstract

In this note, we investigate the Cauchy problem for Keller–Segel system with fractional diffusion for the initial data $(u_0,v_0)$ in the critical Fourier–Bessov–Morrey spaces $\mathcal{FN}_{q,\mu,r}^{2-2\alpha+d-\frac{d-\mu}{q}}(\mathbb R^d)\times \mathcal{FN}_{q,\mu,r}^{2-\alpha+d-\frac{d-\mu}{q}}(\mathbb R^d)$ with $1 < \alpha\le 2$. The global well-posedness with a small initial data of the solution to Keller–Segel system of double-parabolic type is established.