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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Long time heat diffusion on homogeneous trees


Authors: Guia Medolla and Alberto G. Setti
Journal: Proc. Amer. Math. Soc. 128 (2000), 1733-1742
MSC (2000): Primary 43A85, 35K05; Secondary 39A12
Published electronically: November 23, 1999
MathSciNet review: 1694874
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Abstract: Let ${\mathfrak{X}}$ be a homogeneous tree of degree $q+1$, $q\geq 2$, ${\mathcal{L}}$ the Laplace operator of ${\mathfrak{X}}$ and $h_{t}(x)$ the fundamental solution of the heat equation $(\partial _{t} +{\mathcal{L}}) u=0$ on $\mathfrak{X}$. We show that the heat kernel $h_{t}(x)$ is asymptotically concentrated in an annulus moving to infinity with finite speed $R_{1}=(q-1)/(q+1)$. Asymptotic concentration of heat in the $L^{p}$ norm is also investigated.


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Additional Information

Guia Medolla
Affiliation: Dipartimento di Matematica, Politecnico di Milano, via Bonardi 9, I-20133 Milano, Italy
Address at time of publication: I.T.I.S Hensemberger, via Berchet, I-20052 Monza, MI, Italy
Email: guimed@iol.it

Alberto G. Setti
Affiliation: Dipartimento di Scienze Chimiche Fisiche e Matematiche, Università dell’Insubria - Polo di Como, via Lucini 3, I-22100 Como, Italy
Email: setti@fis.unico.it

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05536-7
PII: S 0002-9939(99)05536-7
Keywords: Homogeneous trees, heat equation, long time heat diffusion
Received by editor(s): July 14, 1998
Published electronically: November 23, 1999
Additional Notes: The first author acknowledges financial support through a post-doctoral fellowship at the Politecnico di Milano.
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society