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STRUCTURAL STABILITY RESULTS FOR THE THERMOELASTICITY OF TYPE III

  • Liu, Yan
  • Received : 2011.08.04
  • Published : 2014.09.30

Abstract

The equations arising from the thermoelastic theory are analyzed in a linear approximation. First, we establish the convergence result on the coefficient c. Next, we establish that the solution depends continuously on changes in the coefficient c. The main tool used in this paper is the energy method.

Keywords

convergence result;continuous dependence;thermoelasticity of type III;structural stability

References

  1. K. A. Ames and B. Straughan, Non-Standard and Improperly Posed Problems, Mathe-matics in Science and Engineering Series, Vol. 194, Academic, Press, San Diego, 1997.
  2. A. O. Celebi, V. K. Kalantarov, and D. Ugurlu, On continuous dependence on coefficients of the Brinkman-Forchheimer equations, Appl. Math. Lett. 19 (2006), no. 8, 801-807. https://doi.org/10.1016/j.aml.2005.11.002
  3. A. O. Celebi, V. K. Kalantarov, and D. Ugurlu, Continuous dependence for the convective Brinkman-Forchheimer equations, Appl. Anal. 84 (2005), no. 9, 877-888. https://doi.org/10.1080/00036810500148911
  4. S. Chirita and R. Quintanilla, Spatial decay estimates of Saint-Venant's type in generalized thermoelasticity, Internat. J. Engng. Sci. 34 (1996), no. 3, 299-311. https://doi.org/10.1016/0020-7225(95)00101-8
  5. F. Franchi and B. Straughan, Continuous dependence and decay for the Forchheimer equations, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459 (2003), no. 2040, 3195-3202. https://doi.org/10.1098/rspa.2003.1169
  6. A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics , Proc. Roy. Soc. London Ser. A 432 (1991), no. 1885, 171-194. https://doi.org/10.1098/rspa.1991.0012
  7. A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity 31 (1993), no. 3, 189-208. https://doi.org/10.1007/BF00044969
  8. C. O. Horgan and L. E. Payne, On inequalities of Korn, Friedrichs and Bobuska-Aziz, Arch. Rational Mech. Anal. 82 (1983), no. 2, 165-179.
  9. Y. Liu and C. Lin, Phragmen-Lindelof alternative and continuous dependence-type results for the thermoelasticity of type III, Appl. Anal. 87 (2008), no. 4, 431-449. https://doi.org/10.1080/00036810801927963
  10. C. Lin and L. E. Payne, Structural stability for a Brinkman fluid, Math. Methods Appl. Sci. 30 (2007), no. 5, 567-578. https://doi.org/10.1002/mma.799
  11. L. E. Payne and J. C. Song, Phragmen-Lindelof and continuous dependence type results in generalized heat conduction, Z. Angew. Math. Phys. 47 (1996), no. 4, 527-538. https://doi.org/10.1007/BF00914869
  12. L. E. Payne, J. C. Song, and B. Straughan, Continuous dependence and convergence results for Brinkman and Forchheimer models with variable viscosity, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999), no. 1986, 2173-2190. https://doi.org/10.1098/rspa.1999.0398
  13. L. E. Payne and B. Straughan, Stability in the initial-time geometry problem for the Brinkman and Darcy equations of flow in porous media, J. Math. Pures Appl. 75 (1996), no. 3, 255-271.
  14. L. E. Payne and B. Straughan, Convergence of the equations for a Maxwell fluid, Stud. Appl.Math. 103 (1999), no. 3, 267-278. https://doi.org/10.1111/1467-9590.00128
  15. L. E. Payne and B. Straughan, Structural stability for the Darcy equations of flow in porous media, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454 (1998), no. 1974, 1691-1698. https://doi.org/10.1098/rspa.1998.0227
  16. R. Quintanilla, Logarithmic convexity in thermoelasticity of type III, Mathematical and Numerical Aspects of Wave Propagation, pp. 192-196, SIAM, Philadelphia, PA, 2000.
  17. R. Quintanilla, Damping of end effects in a thermoelastic theory, Appl. Math. Lett. 14 (2001), no. 2, 137-141. https://doi.org/10.1016/S0893-9659(00)00125-7
  18. R. Quintanilla, Convergence and structural stability in thermoelasticity, Appl. Math. Comput. 135 (2003), no. 2-3, 287-300. https://doi.org/10.1016/S0096-3003(01)00331-9
  19. R. Quintanilla, On the spatial behavior of constrained motion in type III thermoelasticity, J. Thermal Stresses 33 (2010), no. 7, 694-705. https://doi.org/10.1080/01495739.2010.482345
  20. J. C. Song, Phragmen-Lindelof and continuous dependence type results in a stokes flow, Appl. Math. Mech. (English Ed.) 31 (2010), no. 7, 875-882. https://doi.org/10.1007/s10483-010-1321-z
  21. B. Straughan, The Energy Method, Stability and Nonlinear Convection, Second Edition, Springer, Appl. Math. Sci. Ser., vol. 91, 2004.
  22. B. Straughan, Stability and Wave Motion in Porous Media, Springer, Appl. Math. Sci. Ser., vol. 165, 2008.
  23. B. Straughan, Explosive Instabilities in Mechanics, Springer, Berlin-heidelberg, 1998.

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