The Keller-Segel system of parabolic-parabolic type in Morrey space

We show the existence of global and local solutions depending on the initial data in the Morrey space to the Keller-Segel system of parabolic-parabolic type. We solve this system by the successive approximation based on the estimation of the heat semigroup in the Morrey space and on the fractional power of the Laplace operator.