Global, decaying solutions of a focusing energy-critical heat equation in ${\mathbb{R}}^4$

We study solutions of the focusing energy-critical nonlinear heat equation $u_t=\mathrm{\Delta }u-\mathrm{\mid }u{\mathrm{\mid }}^{2}u$ in ${\mathbb{R}}^4\mathrm{.}$ 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.