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October  2020, 19(10): 4879-4898. doi: 10.3934/cpaa.2020216

Stability of non-classical thermoelasticity mixture problems

Department of Mathematics, Federal University of Viçosa, Viçosa, MG, 36570-000, Brazil

* Corresponding author

Received  November 2019 Revised  May 2020 Published  July 2020

Fund Project: This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior Brasil (CAPES) - Finance Code 001, and by Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG).

We discuss the stability problem for binary mixtures systems coupled with heat equations. The present manuscript covers the non-classical thermoelastic theories of Coleman-Gurtin and Gurtin-Pipkin - both theories overcome the property of infinite propagation speed (Fourier's law property). We first state the well-posedness and our main result is related to long-time behavior. More precisely, we show, under suitable hypotheses on the physical parameters, that the corresponding solution is stabilized to zero with exponential or rational rates.

Citation: Margareth S. Alves, Rodrigo N. Monteiro. Stability of non-classical thermoelasticity mixture problems. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4879-4898. doi: 10.3934/cpaa.2020216
References:
[1]

J. E. Adkins, Non-linear diffusion, I. Diffusion and flow of mixtures of fluids, Philos. Trans. Roy. Soc. London A, 255 (1963), 607-633.  doi: 10.1098/rsta.1963.0013.

[2]

J. E. Adkins and R. E. Craine, Continuum Theories of Mixtures: Applications, IMA J. Appl. Math., 17 (1976), 153-207. 

[3]

M. S. AlvesJ. E. Muñoz Rivera and R. Quintanilla, Exponential decay in a thermoelastic mixture of solids, Int. J. Solids Struct., 46 (2009), 1659-1666.  doi: 10.1016/j.ijsolstr.2008.12.005.

[4]

M. S. AlvesJ. E. Muñoz RiveraM. Sepúlveda and O. V. Villagrán, Exponential stability in thermoviscoelastic mixtures of solids, Int. J. Solids Struct., 46 (2009), 4151-4162. 

[5]

M. S. AlvesJ. E. Muñoz RiveraM. Sepúlveda and O. V. Villagrán, Analyticity of semigroups associated with thermoviscoelastic mixtures of solids, J. Therm. Stress., 32 (2009), 986-1004. 

[6]

M. S. AlvesM. V. FerreiraJ. E. Muñoz Rivera and O. V. Villagrán, Stability of a thermoelastic mixtures with second sound, Math. Mech. Solids, 24 (2019), 1692-1706.  doi: 10.1177/1081286518775794.

[7]

A. Bedford and D. S. Drumheller, Recent advances. Theories of immiscible and structured mixtures, Int. J. Engng. Sci., 21 (1983), 863-960.  doi: 10.1016/0020-7225(83)90071-X.

[8]

A. Bedford and M. A. Stern, A multi-continuum theory for composite elastic materials, Acta Mechanica, 14 (1972), 85-102. 

[9]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2009), 455-478.  doi: 10.1007/s00208-009-0439-0.

[10]

M. Coti ZelatiF. Dell'Oro and V. Pata, Energy decay of type III linear thermoelastic plates with memory, J. Math. Anal. Appl., 401 (2013), 357-366.  doi: 10.1016/j.jmaa.2012.12.031.

[11]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.

[12]

S. ElangovanB. S. Altan and G. M. Odegard, An elastic micropolar mixture theory for predicting elastic properties of cellular materials, Mech. Mater., 40 (2008), 602-615. 

[13]

J. R. FernándezA. Magaña.M. Masid and R. Quintanilla, On the Viscoelastic Mixtures of Solids, Appl. Math. Optim., 79 (2019), 309-326.  doi: 10.1007/s00245-017-9439-8.

[14]

H. R. Gouin, Variational Theory of Mixtures in Continuum Mechanics, Eur. J. of Mech. B Fluids, 9 (1990), 469-491. 

[15]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, in Evolution Equations, Semigroups and Functional Analysis (eds. A. Lorenzi and B. Ruf), Birkhäuser, Basel, (2002), 155–178.

[16]

M. GrasselliJ. E. Muñoz Rivera and V. Pata, On the energy decay of the linear thermoelastic plate with memory, J. Math. Anal. Appl., 309 (2005), 1-14.  doi: 10.1016/j.jmaa.2004.10.071.

[17]

D. Ieşan and R. Quintanilla, A theory of porous thermoviscoelastic mixtures, J. Thermal Stresses, 30 (2007), 693-714.  doi: 10.1080/01495730701212880.

[18]

S. M. Klisch and J. C. Lot, A special theory of biphasic mixtures and experimental results for human annulus fibrosus tested in confined compression, J. Biomech. Eng., 122 (2000), 180-188. 

[19]

Z. Liu and S. Zheng, Semigroups Associated to Dissipative Systems, Chapman & Hall/CRC Boca Raton, 1999.

[20]

F. Martinez and R. Quintanilla, Some qualitative results for the linear theory of binary mixtures of thermoelastic solids, Collect. Math., 46 (1995), 263-277. 

[21]

J. E. Muñoz RiveraM. G. Naso and R. Quintanilla, Decay of solutions for a mixture of thermoelastic one dimensional solids, Comput. Math. Appl., 66 (2013), 41-55.  doi: 10.1016/j.camwa.2013.03.022.

[22]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol. 44, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[23]

J. Pruss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.

[24]

R. Quintanilla, Existence and exponential decay in the linear theory of viscoelastic mixtures, Eur. J. Mech. A Solids, 24 (2005), 311-324.  doi: 10.1016/j.euromechsol.2004.11.008.

[25]

C. Truesdell, In Continuum Mechanics II: The Rational Mechanics of Materials, Gordon & Breach, New York, 1965.

show all references

References:
[1]

J. E. Adkins, Non-linear diffusion, I. Diffusion and flow of mixtures of fluids, Philos. Trans. Roy. Soc. London A, 255 (1963), 607-633.  doi: 10.1098/rsta.1963.0013.

[2]

J. E. Adkins and R. E. Craine, Continuum Theories of Mixtures: Applications, IMA J. Appl. Math., 17 (1976), 153-207. 

[3]

M. S. AlvesJ. E. Muñoz Rivera and R. Quintanilla, Exponential decay in a thermoelastic mixture of solids, Int. J. Solids Struct., 46 (2009), 1659-1666.  doi: 10.1016/j.ijsolstr.2008.12.005.

[4]

M. S. AlvesJ. E. Muñoz RiveraM. Sepúlveda and O. V. Villagrán, Exponential stability in thermoviscoelastic mixtures of solids, Int. J. Solids Struct., 46 (2009), 4151-4162. 

[5]

M. S. AlvesJ. E. Muñoz RiveraM. Sepúlveda and O. V. Villagrán, Analyticity of semigroups associated with thermoviscoelastic mixtures of solids, J. Therm. Stress., 32 (2009), 986-1004. 

[6]

M. S. AlvesM. V. FerreiraJ. E. Muñoz Rivera and O. V. Villagrán, Stability of a thermoelastic mixtures with second sound, Math. Mech. Solids, 24 (2019), 1692-1706.  doi: 10.1177/1081286518775794.

[7]

A. Bedford and D. S. Drumheller, Recent advances. Theories of immiscible and structured mixtures, Int. J. Engng. Sci., 21 (1983), 863-960.  doi: 10.1016/0020-7225(83)90071-X.

[8]

A. Bedford and M. A. Stern, A multi-continuum theory for composite elastic materials, Acta Mechanica, 14 (1972), 85-102. 

[9]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2009), 455-478.  doi: 10.1007/s00208-009-0439-0.

[10]

M. Coti ZelatiF. Dell'Oro and V. Pata, Energy decay of type III linear thermoelastic plates with memory, J. Math. Anal. Appl., 401 (2013), 357-366.  doi: 10.1016/j.jmaa.2012.12.031.

[11]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.

[12]

S. ElangovanB. S. Altan and G. M. Odegard, An elastic micropolar mixture theory for predicting elastic properties of cellular materials, Mech. Mater., 40 (2008), 602-615. 

[13]

J. R. FernándezA. Magaña.M. Masid and R. Quintanilla, On the Viscoelastic Mixtures of Solids, Appl. Math. Optim., 79 (2019), 309-326.  doi: 10.1007/s00245-017-9439-8.

[14]

H. R. Gouin, Variational Theory of Mixtures in Continuum Mechanics, Eur. J. of Mech. B Fluids, 9 (1990), 469-491. 

[15]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, in Evolution Equations, Semigroups and Functional Analysis (eds. A. Lorenzi and B. Ruf), Birkhäuser, Basel, (2002), 155–178.

[16]

M. GrasselliJ. E. Muñoz Rivera and V. Pata, On the energy decay of the linear thermoelastic plate with memory, J. Math. Anal. Appl., 309 (2005), 1-14.  doi: 10.1016/j.jmaa.2004.10.071.

[17]

D. Ieşan and R. Quintanilla, A theory of porous thermoviscoelastic mixtures, J. Thermal Stresses, 30 (2007), 693-714.  doi: 10.1080/01495730701212880.

[18]

S. M. Klisch and J. C. Lot, A special theory of biphasic mixtures and experimental results for human annulus fibrosus tested in confined compression, J. Biomech. Eng., 122 (2000), 180-188. 

[19]

Z. Liu and S. Zheng, Semigroups Associated to Dissipative Systems, Chapman & Hall/CRC Boca Raton, 1999.

[20]

F. Martinez and R. Quintanilla, Some qualitative results for the linear theory of binary mixtures of thermoelastic solids, Collect. Math., 46 (1995), 263-277. 

[21]

J. E. Muñoz RiveraM. G. Naso and R. Quintanilla, Decay of solutions for a mixture of thermoelastic one dimensional solids, Comput. Math. Appl., 66 (2013), 41-55.  doi: 10.1016/j.camwa.2013.03.022.

[22]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol. 44, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[23]

J. Pruss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.

[24]

R. Quintanilla, Existence and exponential decay in the linear theory of viscoelastic mixtures, Eur. J. Mech. A Solids, 24 (2005), 311-324.  doi: 10.1016/j.euromechsol.2004.11.008.

[25]

C. Truesdell, In Continuum Mechanics II: The Rational Mechanics of Materials, Gordon & Breach, New York, 1965.

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