Global, decaying solutions of a focusing energy-critical heat equation in R4

We study solutions of the focusing energy-critical nonlinear heat equation ut=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 [33], [34]. First, global such 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-Sverak [17], [18] in an argument similar to that of Kenig-Koch [32] for the Navier-Stokes equations.