Decay rates for the 4D energy-critical nonlinear heat equation

In this paper we address the decay of solutions to the four-dimen-sional energy-critical nonlinear heat equation in the critical space ${\dot{H}}^1$. Recently, it was proven that the ${\dot{H}}^1$ norm of solutions goes to zero when time goes to infinity, but no decay rates were established. By means of the Fourier Splitting Method and using properties arising from the scale invariance, we obtain an algebraic upper bound for the decay rate of solutions.