Evolution Equations and Control Theory (EECT)

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

Pages: 241 - 263, Volume 4, Issue 3, September 2015      doi:10.3934/eect.2015.4.241

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Moncef Aouadi - Ecole Nationale d'Ingénieurs de Bizerte, Université de Carthage, BP66, Campus Universitaire Menzel Abderrahman 7035, Tunisia (email)
Alain Miranville - Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348 - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, 86962 Chasseneuil Futuroscope Cedex, France (email)

Abstract: 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:  Thermoelastic diffusion plate, memory, global solutions, global attractor.
Mathematics Subject Classification:  Primary: 35B40, 35B41; Secondary: 35A01, 35B35.

Received: February 2015;      Revised: April 2015;      Available Online: September 2015.