Well-posedness of the Keller-Segel system in Fourier-Besov-Morrey spaces (Q1624156)

Summary: 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 \(1 < \alpha\leq 2\). The global well-posedness with a small initial data of the solution to Keller-Segel system of double-parabolic type is established.