Title: Asymptotic behaviour methods for the Heat Equation. Convergence to the Gaussian,
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.