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

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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

Abstract
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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. doi: 10.1007/s10483-015-1930-7.
[2]	M. Aouadi, On uniform decay of a nonsimple thermoelastic bar with memory,, J. Math. Anal. Appl., 402* (2013), 745. doi: 10.1016/j.jmaa.2013.01.059.*
[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. doi: 10.1002/zamm.201300050.
[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. doi: 10.1016/j.jmaa.2014.02.042.*
[5]	I. Chueshov and I. Lasiecka, Attractors for second order evolution equations with a nonlinear damping,, J. Dyn. Diff. Eq., 16 (2004), 469. doi: 10.1007/s10884-004-4289-x.
[6]	I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,, Mem. Amer. Math. Soc., 195* (2008). doi: 10.1090/memo/0912.*
[7]	I. Chueshov and I. Lasiecka, Von Karman Evolution Equations,, Springer Monographs in Mathematics, (2010). doi: 10.1007/978-0-387-87712-9.
[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. doi: 10.1016/j.jmaa.2012.12.031.*
[9]	T. Fastovska, Upper semicontinuous attractor for 2D Mindlin-Timoshenko thermoelastic model with memory,, Commun. Pure Appl. Anal., 6 (2007), 83. doi: 10.3934/cpaa.2007.6.83.
[10]	T. Fastovska, Upper semicontinuous attractor for a 2D Mindlin-Timoshenko thermo-viscoelastic model with memory,, Nonlinear Analysis TMA, 71 (2009), 4833. doi: 10.1016/j.na.2009.03.059.
[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. doi: 10.1007/s000300050049.
[12]	C. Giorgi and V. Pata, Stability of abstract linear thermoelastic systems with memory,, Math. Models Meth. Appl. Sci., 11 (2001), 627. doi: 10.1142/S0218202501001021.
[13]	M. Grasselli and V. Pata, Uniform attractors of nonautonomous systems with memory,, in Evolution Equations, (2002), 155.
[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. doi: 10.1016/j.jmaa.2004.10.071.*
[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.
[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. doi: 10.1007/BF00281373.
[17]	J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Mathematical Surveys and Monographs, (1988).
[18]	O. Ladyzhenskaya, Attractors for Semi-groups and Evolution Equations,, Cambridge University Press, (1991). doi: 10.1017/CBO9780511569418.
[19]	A. Miranville and S. Zelik, Handbook of Differential Equations, Evolutionary Equations, Vol. 4,, Chapter 3 (eds. C. M. Dafermos and M. Pokorny), (2008).
[20]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1.
[21]	M. Potomkin, Asymptotic behavior of thermoviscoelastic Berger plate,, Commun. Pure Appl. Anal., 9 (2010), 161. doi: 10.3934/cpaa.2010.9.161.
[22]	M. Potomkin, A nonlinear transmission problem for a compound plate with thermoelastic part,, Math. Meth. Appl. Sci., 35 (2012), 530. doi: 10.1002/mma.1589.
[23]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Applied Mathematical Sciences, (1988). doi: 10.1007/978-1-4684-0313-8.
[24]	H. Wu, Long-time behavior for a nonlinear plate equation with thermal memory,, J. Math. Anal. Appl., 348* (2008), 650. doi: 10.1016/j.jmaa.2008.08.001.*
[25]	S. Zheng, Nonlinear Evolution Equations,, Pitman Ser. Monogr. Surv. Pure Appl. Math., (2004). doi: 10.1201/9780203492222.

show all references

References:

[1]	M. Aouadi, On thermoelastic diffusion thin plates theory,, Appl. Math. Mech., 36 (2015), 619. doi: 10.1007/s10483-015-1930-7.
[2]	M. Aouadi, On uniform decay of a nonsimple thermoelastic bar with memory,, J. Math. Anal. Appl., 402* (2013), 745. doi: 10.1016/j.jmaa.2013.01.059.*
[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. doi: 10.1002/zamm.201300050.
[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. doi: 10.1016/j.jmaa.2014.02.042.*
[5]	I. Chueshov and I. Lasiecka, Attractors for second order evolution equations with a nonlinear damping,, J. Dyn. Diff. Eq., 16 (2004), 469. doi: 10.1007/s10884-004-4289-x.
[6]	I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,, Mem. Amer. Math. Soc., 195* (2008). doi: 10.1090/memo/0912.*
[7]	I. Chueshov and I. Lasiecka, Von Karman Evolution Equations,, Springer Monographs in Mathematics, (2010). doi: 10.1007/978-0-387-87712-9.
[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. doi: 10.1016/j.jmaa.2012.12.031.*
[9]	T. Fastovska, Upper semicontinuous attractor for 2D Mindlin-Timoshenko thermoelastic model with memory,, Commun. Pure Appl. Anal., 6 (2007), 83. doi: 10.3934/cpaa.2007.6.83.
[10]	T. Fastovska, Upper semicontinuous attractor for a 2D Mindlin-Timoshenko thermo-viscoelastic model with memory,, Nonlinear Analysis TMA, 71 (2009), 4833. doi: 10.1016/j.na.2009.03.059.
[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. doi: 10.1007/s000300050049.
[12]	C. Giorgi and V. Pata, Stability of abstract linear thermoelastic systems with memory,, Math. Models Meth. Appl. Sci., 11 (2001), 627. doi: 10.1142/S0218202501001021.
[13]	M. Grasselli and V. Pata, Uniform attractors of nonautonomous systems with memory,, in Evolution Equations, (2002), 155.
[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. doi: 10.1016/j.jmaa.2004.10.071.*
[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.
[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. doi: 10.1007/BF00281373.
[17]	J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Mathematical Surveys and Monographs, (1988).
[18]	O. Ladyzhenskaya, Attractors for Semi-groups and Evolution Equations,, Cambridge University Press, (1991). doi: 10.1017/CBO9780511569418.
[19]	A. Miranville and S. Zelik, Handbook of Differential Equations, Evolutionary Equations, Vol. 4,, Chapter 3 (eds. C. M. Dafermos and M. Pokorny), (2008).
[20]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1.
[21]	M. Potomkin, Asymptotic behavior of thermoviscoelastic Berger plate,, Commun. Pure Appl. Anal., 9 (2010), 161. doi: 10.3934/cpaa.2010.9.161.
[22]	M. Potomkin, A nonlinear transmission problem for a compound plate with thermoelastic part,, Math. Meth. Appl. Sci., 35 (2012), 530. doi: 10.1002/mma.1589.
[23]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Applied Mathematical Sciences, (1988). doi: 10.1007/978-1-4684-0313-8.
[24]	H. Wu, Long-time behavior for a nonlinear plate equation with thermal memory,, J. Math. Anal. Appl., 348* (2008), 650. doi: 10.1016/j.jmaa.2008.08.001.*
[25]	S. Zheng, Nonlinear Evolution Equations,, Pitman Ser. Monogr. Surv. Pure Appl. Math., (2004). doi: 10.1201/9780203492222.

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