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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 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), Elsevier, 2008.
[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.
[21]	M. Potomkin, Asymptotic behavior of thermoviscoelastic Berger plate, Commun. Pure Appl. Anal., 9 (2010), 161-192. 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), 530-546. doi: 10.1002/mma.1589.
[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.
[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.
[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.

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References:

[1]	M. Aouadi, On thermoelastic diffusion thin plates theory, Appl. Math. Mech., 36 (2015), 619-632. 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), 745-757. 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), 837-852. 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), 143-165. 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), 469-512. 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), viii+183 pp. doi: 10.1090/memo/0912.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 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), Elsevier, 2008.
[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.
[21]	M. Potomkin, Asymptotic behavior of thermoviscoelastic Berger plate, Commun. Pure Appl. Anal., 9 (2010), 161-192. 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), 530-546. doi: 10.1002/mma.1589.
[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.
[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.
[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.

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