Existence of a separating norm for the semilinear heat equation

We consider the initial value problem of the semilinear heat equation in the Fujita supercritical range for which global-in-time solutions may exist. The main result shows the existence of a separating norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_X$ in the sense that, for nonnegative initial data $u_0\in L^{\mathrm{\infty }}({\mathbf{R}}^n)$, the smallness of $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_X$ implies global-in-time existence, whereas the largeness of $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_X$ implies finite-time blow-up. The separating norm is unique up to equivalence on nonnegative data.