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September  2015, 4(3): 241-263. doi: 10.3934/eect.2015.4.241

Quasi-stability and global attractor in nonlinear thermoelastic diffusion plate with memory

Moncef Aouadi 1, and  Alain Miranville 2,

1. 

Ecole Nationale d'Ingénieurs de Bizerte, Université de Carthage, BP66, Campus Universitaire Menzel Abderrahman 7035, Tunisia
2. 

Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348 - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, 86962 Chasseneuil Futuroscope Cedex

Received  February 2015 Revised  April 2015 Published  September 2015

We analyse the longterm properties of a $C_0-$semigroup describing the solutions to a nonlinear thermoelastic diffusion plate, recently derived by Aouadi [1], where the heat and diffusion flux depends on the past history of the temperature and the chemical potential gradients through memory kernels. First we prove the well-posedness of the initial-boundary-value problem using the $C_0-$semigroup theory of linear operators. Then we show, without rotational inertia, that the thermal and chemical potential coupling is strong enough to guarantee the quasi-stability. By showing that the system is gradient and asymptotically compact, the existence of a global attractor whose fractal dimension is finite is proved.
Keywords: global attractor., Thermoelastic diffusion plate, memory, global solutions.
Mathematics Subject Classification: Primary: 35B40, 35B41; Secondary: 35A01, 35B3.
Citation: Moncef Aouadi, Alain Miranville. Quasi-stability and global attractor in nonlinear thermoelastic diffusion plate with memory. Evolution Equations & Control Theory, 2015, 4 (3) : 241-263. doi: 10.3934/eect.2015.4.241
References:
[1]

M. Aouadi, On thermoelastic diffusion thin plates theory, Appl. Math. Mech., 36 (2015), 619-632. doi: 10.1007/s10483-015-1930-7.  Google Scholar

[2]

M. Aouadi, On uniform decay of a nonsimple thermoelastic bar with memory, J. Math. Anal. Appl., 402 (2013), 745-757. doi: 10.1016/j.jmaa.2013.01.059.  Google Scholar

[3]

M. Aouadi, B. Lazzari and R. Nibbi, A theory of thermoelasticity with diffusion under Green-Naghdi models, ZAMM. Z. Angew. Math. Mech., 94 (2014), 837-852. doi: 10.1002/zamm.201300050.  Google Scholar

[4]

A. Barbosa and T. Fu Ma, Long-time dynamics of an extensible plate equation with thermal memory, J. Math. Anal. Appl., 416 (2014), 143-165. doi: 10.1016/j.jmaa.2014.02.042.  Google Scholar

[5]

I. Chueshov and I. Lasiecka, Attractors for second order evolution equations with a nonlinear damping, J. Dyn. Diff. Eq., 16 (2004), 469-512. doi: 10.1007/s10884-004-4289-x.  Google Scholar

[6]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. doi: 10.1090/memo/0912.  Google Scholar

[7]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.  Google Scholar

[8]

M. Coti Zelati, F. 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

[9]

T. Fastovska, Upper semicontinuous attractor for 2D Mindlin-Timoshenko thermoelastic model with memory, Commun. Pure Appl. Anal., 6 (2007), 83-101. doi: 10.3934/cpaa.2007.6.83.  Google Scholar

[10]

T. Fastovska, Upper semicontinuous attractor for a 2D Mindlin-Timoshenko thermo-viscoelastic model with memory, Nonlinear Analysis TMA, 71 (2009), 4833-4851. doi: 10.1016/j.na.2009.03.059.  Google Scholar

[11]

C. Giorgi, A. Marzocchi and V. Pata, Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA Nonlinear Diff. Eq. Appl., 5 (1998), 333-354. doi: 10.1007/s000300050049.  Google Scholar

[12]

C. Giorgi and V. Pata, Stability of abstract linear thermoelastic systems with memory, Math. Models Meth. Appl. Sci., 11 (2001), 627-644. doi: 10.1142/S0218202501001021.  Google Scholar

[13]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous systems with memory, in Evolution Equations, Semigroups and Functional Analysis (eds. A. Lorenzi and B. Ruf), Progr. Nonlinear Differential Equations Appl., 50, Birkhäuser, Boston, 2002, 155-178.  Google Scholar

[14]

M. Grasselli, J. E. Munoz 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

[15]

M. Grasselli and M. Squassina, Exponential stability and singular limit for a linear thermoelastic plate with memory effects, Adv. Math. Sci. Appl., 16 (2006), 15-31.  Google Scholar

[16]

M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Ration. Mech. Anal., 31 (1968), 113-126. doi: 10.1007/BF00281373.  Google Scholar

[17]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988.  Google Scholar

[18]

O. Ladyzhenskaya, Attractors for Semi-groups and Evolution Equations, Cambridge University Press, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar

[19]

A. Miranville and S. Zelik, Handbook of Differential Equations, Evolutionary Equations, Vol. 4, Chapter 3 (eds. C. M. Dafermos and M. Pokorny), Elsevier, 2008.  Google Scholar

[20]

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

[21]

M. Potomkin, Asymptotic behavior of thermoviscoelastic Berger plate, Commun. Pure Appl. Anal., 9 (2010), 161-192. doi: 10.3934/cpaa.2010.9.161.  Google Scholar

[22]

M. Potomkin, A nonlinear transmission problem for a compound plate with thermoelastic part, Math. Meth. Appl. Sci., 35 (2012), 530-546. doi: 10.1002/mma.1589.  Google Scholar

[23]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[24]

H. Wu, Long-time behavior for a nonlinear plate equation with thermal memory, J. Math. Anal. Appl., 348 (2008), 650-670. doi: 10.1016/j.jmaa.2008.08.001.  Google Scholar

[25]

S. Zheng, Nonlinear Evolution Equations, Pitman Ser. Monogr. Surv. Pure Appl. Math., Vol. 133, Chapman & Hall/ CRC Press, Boca Raton, FL, 2004. doi: 10.1201/9780203492222.  Google Scholar

show all references

References:
[1]

M. Aouadi, On thermoelastic diffusion thin plates theory, Appl. Math. Mech., 36 (2015), 619-632. doi: 10.1007/s10483-015-1930-7.  Google Scholar

[2]

M. Aouadi, On uniform decay of a nonsimple thermoelastic bar with memory, J. Math. Anal. Appl., 402 (2013), 745-757. doi: 10.1016/j.jmaa.2013.01.059.  Google Scholar

[3]

M. Aouadi, B. Lazzari and R. Nibbi, A theory of thermoelasticity with diffusion under Green-Naghdi models, ZAMM. Z. Angew. Math. Mech., 94 (2014), 837-852. doi: 10.1002/zamm.201300050.  Google Scholar

[4]

A. Barbosa and T. Fu Ma, Long-time dynamics of an extensible plate equation with thermal memory, J. Math. Anal. Appl., 416 (2014), 143-165. doi: 10.1016/j.jmaa.2014.02.042.  Google Scholar

[5]

I. Chueshov and I. Lasiecka, Attractors for second order evolution equations with a nonlinear damping, J. Dyn. Diff. Eq., 16 (2004), 469-512. doi: 10.1007/s10884-004-4289-x.  Google Scholar

[6]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. doi: 10.1090/memo/0912.  Google Scholar

[7]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.  Google Scholar

[8]

M. Coti Zelati, F. 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

[9]

T. Fastovska, Upper semicontinuous attractor for 2D Mindlin-Timoshenko thermoelastic model with memory, Commun. Pure Appl. Anal., 6 (2007), 83-101. doi: 10.3934/cpaa.2007.6.83.  Google Scholar

[10]

T. Fastovska, Upper semicontinuous attractor for a 2D Mindlin-Timoshenko thermo-viscoelastic model with memory, Nonlinear Analysis TMA, 71 (2009), 4833-4851. doi: 10.1016/j.na.2009.03.059.  Google Scholar

[11]

C. Giorgi, A. Marzocchi and V. Pata, Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA Nonlinear Diff. Eq. Appl., 5 (1998), 333-354. doi: 10.1007/s000300050049.  Google Scholar

[12]

C. Giorgi and V. Pata, Stability of abstract linear thermoelastic systems with memory, Math. Models Meth. Appl. Sci., 11 (2001), 627-644. doi: 10.1142/S0218202501001021.  Google Scholar

[13]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous systems with memory, in Evolution Equations, Semigroups and Functional Analysis (eds. A. Lorenzi and B. Ruf), Progr. Nonlinear Differential Equations Appl., 50, Birkhäuser, Boston, 2002, 155-178.  Google Scholar

[14]

M. Grasselli, J. E. Munoz 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

[15]

M. Grasselli and M. Squassina, Exponential stability and singular limit for a linear thermoelastic plate with memory effects, Adv. Math. Sci. Appl., 16 (2006), 15-31.  Google Scholar

[16]

M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Ration. Mech. Anal., 31 (1968), 113-126. doi: 10.1007/BF00281373.  Google Scholar

[17]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988.  Google Scholar

[18]

O. Ladyzhenskaya, Attractors for Semi-groups and Evolution Equations, Cambridge University Press, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar

[19]

A. Miranville and S. Zelik, Handbook of Differential Equations, Evolutionary Equations, Vol. 4, Chapter 3 (eds. C. M. Dafermos and M. Pokorny), Elsevier, 2008.  Google Scholar

[20]

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

[21]

M. Potomkin, Asymptotic behavior of thermoviscoelastic Berger plate, Commun. Pure Appl. Anal., 9 (2010), 161-192. doi: 10.3934/cpaa.2010.9.161.  Google Scholar

[22]

M. Potomkin, A nonlinear transmission problem for a compound plate with thermoelastic part, Math. Meth. Appl. Sci., 35 (2012), 530-546. doi: 10.1002/mma.1589.  Google Scholar

[23]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[24]

H. Wu, Long-time behavior for a nonlinear plate equation with thermal memory, J. Math. Anal. Appl., 348 (2008), 650-670. doi: 10.1016/j.jmaa.2008.08.001.  Google Scholar

[25]

S. Zheng, Nonlinear Evolution Equations, Pitman Ser. Monogr. Surv. Pure Appl. Math., Vol. 133, Chapman & Hall/ CRC Press, Boca Raton, FL, 2004. doi: 10.1201/9780203492222.  Google Scholar

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