  We study solutions of the focusing energy-critical nonlinear heat equation
$u_t = \Delta u - |u|^2u$ in $\mathbb{R}^4.$ We show that solutions emanating
from initial data with energy and $\dot{H}^1-$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 $L^2$-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.
