Existence, uniqueness and exponential decay: An evolution problem in heat conduction with memory

Carillo, Sandra

doi:10.1090/S0033-569X-2011-01223-1

Author: Sandra Carillo
Journal: Quart. Appl. Math. 69 (2011), 635-649
MSC (2000): Primary 80A20, 74F05
DOI: https://doi.org/10.1090/S0033-569X-2011-01223-1
Published electronically: July 7, 2011
MathSciNet review: 2893993
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Abstract: A rigid linear heat conductor with memory conductor is considered. An evolution problem which arises in studying the thermodynamical state of the material with memory is considered. Specifically, the time evolution of the temperature distribution within a rigid heat conductor with memory is investigated. The constitutive equations which characterize heat conduction with memory involve an integral term since the temperature’s time derivative is connected to the heat flux gradient. The integro-differential problem, when initial and boundary conditions are assigned, is studied to obtain existence and uniqueness results. Key tools turn out to be represented by suitable expressions of the minimum free energy which allow us to construct functional spaces meaningful under both the physical as well as the analytic viewpoint since therein the existence and uniqueness results can be established. Finally, conditions which guarantee exponential decay at infinity are obtained.

References

Mark J. Ablowitz and Athanassios S. Fokas, Complex variables: introduction and applications, 2nd ed., Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2003. MR 1989049
G. Amendola and S. Carillo, Thermal work and minimum free energy in a heat conductor with memory, Quart. J. Mech. Appl. Math. 57 (2004), no. 3, 429–446. MR 2088844, DOI https://doi.org/10.1093/qjmam/57.3.429
Carlo Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena 3 (1949), 83–101 (Italian). MR 0032898
Sandra Carillo, Some remarks on materials with memory: heat conduction and viscoelasticity, J. Nonlinear Math. Phys. 12 (2005), no. suppl. 1, 163–178. MR 2117178, DOI https://doi.org/10.2991/jnmp.2005.12.s1.14
Bernard D. Coleman, Thermodynamics of materials with memory, Arch. Rational Mech. Anal. 17 (1964), 1–46. MR 171419, DOI https://doi.org/10.1007/BF00283864
Bernard D. Coleman and Ellis H. Dill, On thermodynamics and the stability of motions of materials with memory, Arch. Rational Mech. Anal. 51 (1973), 1–53. MR 342046, DOI https://doi.org/10.1007/BF00275991
Richard Datko, Extending a theorem of A. M. Liapunov to Hilbert space, J. Math. Anal. Appl. 32 (1970), 610–616. MR 268717, DOI https://doi.org/10.1016/0022-247X%2870%2990283-0
M. Fabrizio, G. Gentili, and D. W. Reynolds, On rigid linear heat conductors with memory, Internat. J. Engrg. Sci. 36 (1998), no. 7-8, 765–782. MR 1629806, DOI https://doi.org/10.1016/S0020-7225%2897%2900123-7
Morton E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31 (1968), no. 2, 113–126. MR 1553521, DOI https://doi.org/10.1007/BF00281373

M. McCarthy, Constitutive equations for thermomechanical materials with memory, Int. J. Eng. Sci. 8 (1970), pp. 467–474.

Sandro Salsa, Partial differential equations in action, Universitext, Springer-Verlag Italia, Milan, 2008. From modelling to theory. MR 2399851

A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486