Long-time dynamics for a fractional piezoelectric system with magnetic effects and Fourier's law

Abstract In this work, we use a variational approach to model vibrations on a piezoelectric beam with fractional damping depending on a parameter ν ∈ ( 0 , 1 / 2 ) . Magnetic and thermal effects are taken into account via the Maxwell's equations and Fourier law, respectively. Existence and uniqueness of solutions of the system is proved by the semigroup theory. The existence of smooth global attractors with finite fractal dimension and the existence of exponential attractors for the associated dynamical system are proved. Finally, the upper-semicontinuity of global attractors as ν → 0 + is shown.

[1]  Q. Qin,et al.  Macro-Micro Theory on Multifield Coupling Behavior of Heterogeneous Materials , 2009 .

[2]  O. Ladyzhenskaya,et al.  Attractors for Semigroups and Evolution Equations , 1991 .

[3]  A. Ozer,et al.  Modeling and Controlling an Active Constrained Layer (ACL) Beam Actuated by Two Voltage Sources With/Without Magnetic Effects , 2015, IEEE Transactions on Automatic Control.

[4]  W. Nowacki,et al.  SOME GENERAL THEOREMS OF THERMOPIEZOELECTRICITY , 1978 .

[5]  Irena Lasiecka,et al.  Exact null controllability of structurally damped and thermo-elastic parabolic models , 1998 .

[6]  Shuping Chen,et al.  Gevrey class semigroups arising from elastic systems with gentle dissipation: the case 0<<\frac12 , 1990 .

[7]  A. Carvalho,et al.  Attractors for Strongly Damped Wave Equations with Critical Nonlinearities , 2002 .

[8]  Kirsten Morris,et al.  Modeling and Stabilizability of Voltage-Actuated Piezoelectric Beams with Magnetic Effects , 2013, SIAM J. Control. Optim..

[9]  M. M. Freitas,et al.  Continuity of non-autonomous attractors for hyperbolic perturbation of parabolic equations , 2018 .

[10]  Zhijian Yang,et al.  Upper semicontinuity of global attractors for a family of semilinear wave equations with gentle dissipation , 2017, Appl. Math. Lett..

[11]  J. M. Rivera,et al.  Rates of decay to weak thermoelastic Bresse system , 2010 .

[12]  J. U. Kim.  On the energy decay of a linear thermoelastic bar and plate , 1992 .

[13]  R. Racke,et al.  Mildly dissipative nonlinear Timoshenko systems—global existence and exponential stability☆ , 2002 .

[14]  To Fu Ma,et al.  Singular Limit and Long-Time Dynamics of Bresse Systems , 2017, SIAM J. Math. Anal..

[15]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[16]  Rate of attraction for a semilinear wave equation with variable coefficients and critical nonlinearities , 2013 .

[17]  M. Reissig.  Structurally damped elastic waves in 2D , 2016 .

[18]  Roberto Triggiani,et al.  PROOF OF EXTENSIONS OF TWO CONJECTURES ON STRUCTURAL DAMPING FOR ELASTIC SYSTEMS , 1989 .

[19]  A. Carvalho,et al.  Fractional Schrödinger equation; solvability and connection with classical Schrödinger equation , 2018 .

[20]  Enrique Zuazua,et al.  Decay Rates for the Three‐Dimensional Linear System of Thermoelasticity , 1999 .

[21]  Roberto Triggiani,et al.  Regularity of some structurally damped problems with point control and with boundary control , 1991 .

[22]  A. Carvalho,et al.  Local well posedness for strongly damped wave equations with critical nonlinearities , 2002, Bulletin of the Australian Mathematical Society.

[23]  Roberto Triggiani,et al.  Characterization of Domains of Fractional Powers of Certain Operators Arising in Elastic Systems and , 1990 .

[24]  R. Racke,et al.  Smoothing properties, decay, and global existence of solutions to nonlinear coupled systems of thermoelastic type , 1995 .

[25]  M. Gurtin,et al.  A general theory of heat conduction with finite wave speeds , 1968 .

[26]  Igor Chueshov,et al.  Von Karman Evolution Equations: Well-posedness and Long Time Dynamics , 2010 .

[27]  A. Carvalho,et al.  Regularity of solutions on the global attractor for a semilinear damped wave equation , 2008 .

[28]  Enrique Zuazua,et al.  Null‐Controllability of a System of Linear Thermoelasticity , 1998 .

[29]  Zhijian Yang,et al.  Attractors and their stability on Boussinesq type equations with gentle dissipation , 2019, Communications on Pure & Applied Analysis.

[30]  Zhijian Yang,et al.  Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous kirchhoff wave models , 2018 .

[31]  Gaël Chevallier,et al.  MICRO-SLIP INDUCED DAMPING IN PLANAR CONTACT UNDER CONSTANT AND UNIFORM NORMAL STRESS , 2010 .

[32]  R. Racke,et al.  Large Solutions and Smoothing Properties for Nonlinear Thermoelastic Systems , 1996 .

[33]  Zhuangyi Liu,et al.  A note on the equations of a thermoelastic plate , 1995 .

[34]  Jack K. Hale,et al.  Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation , 1988 .

[35]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[36]  R. Quintanilla,et al.  Exponential decay in a thermoelastic mixture of solids , 2009 .

[37]  N. Laskin.  Fractional Schrödinger equation. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  Ralph C. Smith.  Smart Material Systems , 2005 .

[39]  Zhuangyi Liu,et al.  Exponential stability and analyticity of abstract linear thermoelastic systems , 1997 .

[40]  Irena Lasiecka,et al.  Analyticity, and lack thereof, of thermo-elastic semigroups , 1998 .

[41]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[42]  William Alan Day,et al.  The Thermodynamics of Simple Materials with Fading Memory , 1972 .

[43]  M. Vishik,et al.  Attractors of Evolution Equations , 1992 .

[44]  J. P. Lasalle,et al.  Dissipative periodic processes , 1971 .

[45]  James C. Robinson.  Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors , 2001 .

[46]  P. C. Y. Lee.  A variational principle for the equations of piezoelectromagnetism in elastic dielectric crystals , 1991 .

[47]  A. Özer.  Stabilization Results for Well-Posed Potential Formulations of a Current-Controlled Piezoelectric Beam and Their Approximations , 2020 .

[48]  Zhijian Yang,et al.  Stability of exponential attractors for a family of semilinear wave equations with gentle dissipation , 2018 .

[49]  A. Ozer.  Further stabilization and exact observability results for voltage-actuated piezoelectric beams with magnetic effects , 2013, 1311.2053.

[50]  A. Carvalho,et al.  Strongly damped wave problems: Bootstrapping and regularity of solutions , 2008 .

[51]  Zhuangyi Liu,et al.  Exponential stability of the Kirchhoff plate with thermal or viscoelastic damping , 1997 .

[52]  J. D. Martins,et al.  Exponential stabilization of fully dynamic and electrostatic piezoelectric beams with delayed distributed damping feedback , 2021 .

[53]  I. Lasiecka,et al.  ON THE ATTRACTOR FOR A SEMILINEAR WAVE EQUATION WITH CRITICAL EXPONENT AND NONLINEAR BOUNDARY DISSIPATION , 2002 .