Morrey spaces and classification of global solutions for a supercritical semilinear heat equation in Rn

As a consequence we also prove that the set of initial data producing global solutions is open in suitable topologies, and we show that the so-called “borderline” global weak solutions blow up in finite time and then become classical again and decay as t→∞. All these results confirm the key role played by the Morrey space M2,4/(p−1) in the understanding of the structure of the set of global solutions for p>pS.