Spatial decay in transient heat conduction for general elongated regions
Authors:
R. J. Knops and R. Quintanilla
Journal:
Quart. Appl. Math. 76 (2018), 611-625
MSC (2010):
Primary 58J35
DOI:
https://doi.org/10.1090/qam/1497
Published electronically:
December 5, 2017
MathSciNet review:
3855824
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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.
References
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- Ramón Quintanilla, Spatial behaviour for quasilinear parabolic equations in cylinders and cones, NoDEA Nonlinear Differential Equations Appl. 5 (1998), no. 2, 137–146. MR 1619029, DOI https://doi.org/10.1007/s000300050038
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References
- Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
- J. H. Bramble and L. E. Payne, Bounds in the Neumann problem for second order uniformly elliptic operators, Pacific J. Math. 12 (1962), 823–833. MR 0146504
- J. H. Bramble and L. E. Payne, Bounds for solutions of second-order elliptic partial differential equations, Contributions to Differential Equations 1 (1963), 95–127. MR 0163049
- Carlo Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena 3 (1949), 83–101 (Italian). MR 0032898
- Carlo Cattaneo, Sur une forme de l’équation de la chaleur éliminant le paradoxe d’une propagation instantanée, C. R. Acad. Sci. Paris 247 (1958), 431–433 (French). MR 0095680
- A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses 15 (1992), no. 2, 253–264. Sixty-fifth Birthday of Bruno A. Boley Symposium, Part 2 (Atlanta, GA, 1991). MR 1175235
- A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity 31 (1993), no. 3, 189–208. MR 1236373
- J. Hersch, L. E. Payne, and M. M. Schiffer, Some inequalities for Stekloff eigenvalues, Arch. Rational Mech. Anal. 57 (1975), 99–114. MR 0387837
- C. O. Horgan, Eigenvalue estimates and the trace theorem, J. Math. Anal. Appl. 69 (1979), no. 1, 231–242. MR 535293
- C. O. Horgan, L. E. Payne, and L. T. Wheeler, Spatial decay estimates in transient heat conduction, Quart. Appl. Math. 42 (1984), no. 1, 119–127. MR 736512
- C. O. Horgan and R. Quintanilla, Spatial decay of transient end effects in functionally graded heat conducting materials, Quart. Appl. Math. 59 (2001), no. 3, 529–542. MR 1848533
- R. J. Knops, Zanaboni’s formulation of Saint-Venant’s principle extended to linear thermo-elasticity, J. Engrg. Math. 95 (2015), 73–86. MR 3428626
- R. J. Knops and L. E. Payne, The effect of a variation in the elastic moduli on Saint-Venant’s principle for a half-cylinder, J. Elasticity 44 (1996), no. 2, 161–182. MR 1420981
- R. J. Knops and Piero Villaggio, Zanaboni’s treatment of Saint-Venant’s principle, Appl. Anal. 91 (2012), no. 2, 345–370. MR 2876758
- James K. Knowles, On the spatial decay of solutions of the heat equation, Z. Angew. Math. Phys. 22 (1971), 1050–1056 (English, with German summary). MR 0298255
- L. E. Payne and H. F. Weinberger, New bounds for solutions of second order elliptic partial differential equations, Pacific J. Math. 8 (1958), 551–573. MR 0104047
- Ramón Quintanilla, Spatial behaviour for quasilinear parabolic equations in cylinders and cones, NoDEA Nonlinear Differential Equations Appl. 5 (1998), no. 2, 137–146. MR 1619029
- V. G. Sigillito, Explicit a priori inequalities with applications to boundary value problems, Pitman Publishing, London-San Francisco, Calif.-Melbourne, 1977. Research Notes in Mathematics, Vol. 13. MR 0499654
- O. Zanaboni, Dimonstrazione generale dei principio del Saint-Venant, Rend.Accad. Lincei 25 (1937), 117–121.
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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
MR Author ID:
143170
Email:
ramon.quintanilla@upc.edu
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
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