Poincaré's inequality and diffusive evolution equations

2007/11/10 by Clayton Bjorland, Bjorland, Clayton, María E. Schonbek +2
Computer Science · Mathematics · #35Q35 #76B03 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #History and Overview (math.HO) #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #math.AP #math.HO #msc:35Q35 #msc:76B03

paper · pdf · doi:10.48550/arxiv.0711.1626

16 pages. Submitted

arxiv created 2007/11/10 · openalex publication_date 2007/11/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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