Local and global solvability for Keller–Segel system in Besov–Morrey spaces

This paper concerns the Cauchy problems for the Keller–Segel system. The local/global existence of the solution is established for initial data in homogeneous Besov–Morrey spaces in the scaling critical case. Unfortunately, Besov–Morrey spaces are not separable in general. So, attention must be paid to the compatibility of initial data. From this viewpoint, a new closed subspace of Besov–Morrey spaces is introduced to consider the local solution.