Global, decaying solutions of a focusing energy-critical heat equation in R^4

We study solutions of the focusing energy-critical nonlinear heat equation u_t =Δu−|u|²u in R⁴. We show that solutions emanating from initial data with energy and H^˙1 − norm below those of the stationary solution W are global and decay to zero, via the "concentration-compactness plus rigidity" strategy of Kenig-Merle. First, such global solutions are shown to dissipate to zero, using a refinement of the small data theory and the L² -dissipation relation. Finite-time blow-up is then ruled out using the backwards-uniqueness of Escauriaza, Seregin and Sverak in an argument similar to that of Kenig and Koch for the Navier-Stokes equations.