The decay of global solutions of a semilinear heat equation

We are interested in the time decay estimates of global solutions of the semilinear parabolic equation $u_t= \Delta u+|u|^{p-1}u$ in $\R^N\times\R^+$, where $p>1$. We find several new sufficient and/or necessary conditions guaranteeing that the solution for $t$ large behaves like the solution of the linear heat equation or has the self-similar decay. We are particularly interested in the behaviour of threshold solutions lying on the borderline between global existence and blow-up.