Global well-posedness for the generalized Keller-Segel system in critical Besov-Morrey spaces with variable exponent

This article is devoted to studying the generalized Keller-Segel system (GKS) in homogeneous variable exponent Besov-Morrey spaces. By making use of the Littlewood-Paley theory and the Chemin mono-norm methods, we obtain, when 1/2< β ≤ 1, a global well-posedness result for GKS system with small initial data in the critical variable exponent Besov-Morrey spaces N_{r(·),q(·),h}^{-2β+ n/q(·)}(R^n) with 1 ≤ r(·) ≤ q(·) < ∞, 1 ≤ h ≤ ∞. In the limit case β = 1/2 , we show the global well-posedness for small initial data in N_{r(·),q(·),1}^{-1+ n/q(·)}(R^n) with 1 ≤ r(·) ≤ q(·) < ∞.