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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Arbitrarily slow approach to limiting behavior


Authors: K. Golden and S. Goldstein
Journal: Proc. Amer. Math. Soc. 112 (1991), 109-119
MSC: Primary 26A12; Secondary 60H10, 82C44
DOI: https://doi.org/10.1090/S0002-9939-1991-1050020-6
MathSciNet review: 1050020
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Abstract: Let $ f({\mathbf{k}},t):{\mathbb{R}^N} \times [0,\infty ) \to \mathbb{R}$ be jointly continuous in $ {\mathbf{k}}$ and $ t$, with $ {\lim _{t \to \infty }}f({\mathbf{k}},t) = F({\mathbf{k}})$ discontinuous for a dense set of $ {\mathbf{k}}$'s. It is proven that there exists a dense set $ \Gamma $ of $ {\mathbf{k}}$'s such that, for $ {\mathbf{k}} \in \Gamma $, $ \vert f({\mathbf{k}},t) - F({\mathbf{k}})\vert$ approaches 0 arbitrarily slowly, i.e., roughly speaking, more slowly than any expressible function $ g(t) \to 0$. This result is applied to diffusion and conduction in quasiperiodic media and yields arbitrarily slow approaches to limiting behavior as time or volume becomes infinite. Such a slow approach is in marked contrast to the power laws widely found for random media, and, in fact, implies that there is no law whatsoever governing the asymptotics.


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DOI: https://doi.org/10.1090/S0002-9939-1991-1050020-6
Article copyright: © Copyright 1991 American Mathematical Society