Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Spatial decay in transient heat conduction for general elongated regions

Authors: R. J. Knops and R. Quintanilla
Journal: Quart. Appl. Math.
MSC (2010): Primary 58J35
DOI: https://doi.org/10.1090/qam/1497
Published electronically: December 5, 2017
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Abstract: Zanaboni's procedure for establishing Saint-Venant's principle is extended to anisotropic homogeneous transient heat conduction on regions that are successively embedded in each other to become indefinitely elongated. No further geometrical restrictions are imposed. The boundary of each region is maintained at zero temperature apart from the common surface of intersection which is heated to the same temperature assumed to be of bounded time variation. Heat sources are absent. Subject to these conditions, the thermal energy, supposed bounded in each region, becomes vanishingly small in those parts of the regions sufficiently remote from the heated common surface. As with the original treatment, the proof involves certain monotone bounded sequences, and does not depend upon differential inequalities or the maximum principle. A definition of an elongated region is presented.

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Additional Information

R. J. Knops
Affiliation: The Maxwell Institute of Mathematical Sciences, and School of Mathematical and Computing Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, Scotland, United Kingdom
Email: r.j.knops@hw.ac.uk

R. Quintanilla
Affiliation: Departamento de Matemáticas, UPC, C.Colòn 11, 08222, Terrassa, Spain
Email: ramon.quintanilla@upc.edu

DOI: https://doi.org/10.1090/qam/1497
Received by editor(s): November 4, 2016
Received by editor(s) in revised form: November 1, 2017
Published electronically: December 5, 2017
Additional Notes: The second author was supported by the project “Análisis Matemático de las Ecuaciones en Derivadas Parciales de la Termomecánica (MTM2013-42004-P)” and the project “Análisis Matemático de los Problemas de la Termomecánica (MTM2016-74934-P)” of the Ministerio Español de Economia y Competitividad
Article copyright: © Copyright 2017 Brown University

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