Asymptotic behaviour methods for the Heat Equation. Convergence to the Gaussian

TLDR: In this paper, the authors discuss the asymptotic behavior of the solutions of the classical heat equation posed in the whole Euclidean space, and present several methods of proof: first, the scaling method, then several versions of the representation method, and finally, the Boltzmann entropy method, coming from kinetic equations.

Abstract: In this expository work we discuss the asymptotic behaviour of the solutions of the classical heat equation posed in the whole Euclidean space. After an introductory review of the main facts on the existence and properties of solutions, we proceed with the proofs of convergence to the Gaussian fundamental solution, a result that holds for all integrable solutions, and represents in the PDE setting the Central Limit Theorem of probability. We present several methods of proof: first, the scaling method. Then several versions of the representation method. This is followed by the functional analysis approach that leads to the famous related equations, Fokker-Planck and Ornstein-Uhlenbeck. The analysis of this connection is also given in rather complete form here. Finally, we present the Boltzmann entropy method, coming from kinetic equations. The different methods are interesting because of the possible extension to prove the asymptotic behaviour or stabilization analysis for more general equations, linear or nonlinear. It all depends a lot on the particular features, and only one or some of the methods work in each case.Other settings of the Heat Equation are briefly discussed in Section 9 and a longer mention of results for different equations is done in Section 10.