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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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.   Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.   Google Scholar

[19]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

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

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.   Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.   Google Scholar

[19]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

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

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