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 | References | Similar Articles | Additional Information

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

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