Morrey spaces norms and criteria for blowup in chemotaxis models

Two-dimensional Keller--Segel models for the chemotaxis with fractional (anomalous) diffusion are considered. Criteria for blowup of solutions in terms of suitable Morrey spaces norms are derived. Similarly, a criterion for blowup of solutions in terms of the radial initial concentrations, related to suitable Morrey spaces norms, is shown for radially symmetric solutions of chemotaxis in several dimensions. Those conditions are, in a sense, complementary to the ones guaranteeing the global-in-time existence of solutions.