Poincar\'e's inequality and diffusive evolution equations

TL;DR

This paper explains the transition from exponential to algebraic decay in diffusive evolution equations by analyzing the spectrum of the Dirichlet Laplacian, and discusses data requirements for different decay behaviors.

Contribution

It provides a spectral analysis framework to understand the change in decay rates in diffusive equations across bounded and unbounded domains.

Findings

Spectral properties determine decay rate transitions.

Bounded domains exhibit exponential decay.

Unbounded domains may show algebraic or no decay.

Abstract

This paper addresses the question of change of decay rate from exponential to algebraic for diffusive evolution equations. We show how the behaviour of the spectrum of the Dirichlet Laplacian in the two cases yields the passage from exponential decay in bounded domains to algebraic decay or no decay at all in the case of unbounded domains. It is well known that such rates of decay exist: the purpose of this paper is to explain what makes the change in decay happen. We also discuss what kind of data is needed to obtain various decay rates.