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December  2012, 1(2): 251-270. doi: 10.3934/eect.2012.1.251

Memory relaxation of type III thermoelastic extensible beams and Berger plates

Filippo Dell'Oro 1, and  Vittorino Pata 2,

1. 

Dipartimento di Matematica "Francesco Brioschi", Politecnico di Milano, Via Bonardi 9, Milano 20133
2. 

Dipartimento di Matematica “Francesco Brioschi”, Politecnico di Milano, Via Bonardi 9, Milano 20133

Received  February 2012 Revised  July 2012 Published  October 2012

We analyze an abstract version of the evolution system ruling the dynamics of a memory relaxation of a type III thermoelastic extensible beam or Berger plate occupying a volume $\Omega$ \begin{equation} \begin{cases} u_{tt}-ωΔ u_{tt}+Δ^2 u-[b +||\nabla u\|^2_{L^2(\Omega)}]\Delta u+Δ α_t=g\\ α_{tt}-Δ α-∫_0^\infty u(s)Δ[α(t)-α(t-s)]d s-Δ u_t=0 \end{cases} \end{equation} subject to hinged boundary conditions for $u$ and to the Dirichlet boundary condition for $\alpha$, where the dissipation is entirely contributed by the convolution term in the second equation. The study of the asymptotic properties of the related solution semigroup is addressed.
Keywords: Type III thermoelastic beams and plates, equations with memory, global attractor, Lyapunov functional, regularity..
Mathematics Subject Classification: Primary: 35B41, 74F05; Secondary: 37L45, 45K0.
Citation: Filippo Dell'Oro, Vittorino Pata. Memory relaxation of type III thermoelastic extensible beams and Berger plates. Evolution Equations and Control Theory, 2012, 1 (2) : 251-270. doi: 10.3934/eect.2012.1.251
References:
[1]

H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472.

[2]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992.

[3]

F. Bucci and I. Chueshov, Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations, Discrete Cont. Dyn. Systems, 22 (2008), 557-586. doi: 10.3934/dcds.2008.22.557.

[4]

V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal., 46 (2006), 251-273.

[5]

C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Phys. Rev. Lett., 94 (2005), p.154301. doi: 10.1103/PhysRevLett.94.154301.

[6]

I. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems," Acta, Kharkov, 2002.

[7]

I. Chueshov and I. Lasiecka, Attractors and long-time behavior of von Karman thermoelastic plates, Appl. Math. Optim., 58 (2008), 195-241. doi: 10.1007/s00245-007-9031-8.

[8]

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.

[9]

M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal., 4 (2005), 705-720.

[10]

M. Coti Zelati, F. Dell'Oro and V. Pata, Energy decay of type III linear thermoelastic plates with memory, submitted.

[11]

M. Coti Zelati, V. Pata and R. Quintanilla, Regular global attractors of type III thermoelastic extensible beams, Chin. Ann. Math. Series B, 31 (2010), 619-630. doi: 10.1007/s11401-010-0605-4.

[12]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.

[13]

G. Fichera, Is the Fourier theory of heat propagation paradoxical?, Rend. Circ. Mat. Palermo, 41 (1992), 5-28.

[14]

S. Gatti, A. Miranville, V. Pata and S. Zelik, Attractors for semilinear equations of viscoelasticity with very low dissipation, Rocky Mountain J. Math., 38 (2008), 1117-1138. doi: 10.1216/RMJ-2008-38-4-1117.

[15]

S. Gatti, A. Miranville, V. Pata and S. Zelik, Continuous families of exponential attractors for singularly perturbed equations with memory, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 329-366. doi: 10.1017/S0308210509000365.

[16]

C. Giorgi, M. G. Naso, V. Pata and M. Potomkin, Global attractors for the extensible thermoelastic beam system, J. Diff. Eqs, 246 (2009), 3496-3517. doi: 10.1016/j.jde.2009.02.020.

[17]

C. Giorgi, V. Pata and E. Vuk, On the extensible viscoelastic beam, Nonlinearity, 21 (2008), 713-733. doi: 10.1088/0951-7715/21/4/004.

[18]

A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses, 15 (1992), 253-264. doi: 10.1080/01495739208946136.

[19]

A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208. doi: 10.1007/BF00044969.

[20]

A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. I. Classical continuum physics, Proc. Roy. Soc. London A, 448 (1995), 335-356. doi: 10.1098/rspa.1995.0020.

[21]

A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. II. Generalized continua, Proc. Roy. Soc. London A, 448 (1995), 357-377. doi: 10.1098/rspa.1995.0021.

[22]

A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. III. Mixtures of interacting continua, Proc. Roy. Soc. London A, 448 (1995), 379-388. doi: 10.1098/rspa.1995.0022.

[23]

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.

[24]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," American Mathematical Society, Providence, 1988.

[25]

A. Haraux, "Syst\`emes Dynamiques Dissipatifs et Applications," Coll. RMA no.17, Masson, Paris, 1991.

[26]

O. A. Ladyzhenskaya, Finding minimal global attractors for the Navier-Stokes equations and other partial differential equations, Russian Math. Surveys, 42 (1987), 27-73. doi: 10.1070/RM1987v042n06ABEH001503.

[27]

V. Pata, Exponential stability in linear viscoelasticity, Quart. Appl. Math., 64 (2006), 499-513.

[28]

V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333-360. doi: 10.1007/s00032-009-0098-3.

[29]

V. Pata, Uniform estimates of Gronwall type, J. Math. Anal. Appl., 373 (2011), 264-270. doi: 10.1016/j.jmaa.2010.07.006.

[30]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.

[31]

M. Potomkin, Asymptotic behavior of thermoviscoelastic Berger plate, Commun. Pure Appl. Anal., 9 (2010), 161-192.

[32]

B. Straughan, "Heat Waves," Springer, New York, 2011.

[33]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," Springer, New York, 1997.

[34]

S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36.

show all references

References:
[1]

H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472.

[2]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992.

[3]

F. Bucci and I. Chueshov, Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations, Discrete Cont. Dyn. Systems, 22 (2008), 557-586. doi: 10.3934/dcds.2008.22.557.

[4]

V. V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal., 46 (2006), 251-273.

[5]

C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving media, Phys. Rev. Lett., 94 (2005), p.154301. doi: 10.1103/PhysRevLett.94.154301.

[6]

I. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems," Acta, Kharkov, 2002.

[7]

I. Chueshov and I. Lasiecka, Attractors and long-time behavior of von Karman thermoelastic plates, Appl. Math. Optim., 58 (2008), 195-241. doi: 10.1007/s00245-007-9031-8.

[8]

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.

[9]

M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal., 4 (2005), 705-720.

[10]

M. Coti Zelati, F. Dell'Oro and V. Pata, Energy decay of type III linear thermoelastic plates with memory, submitted.

[11]

M. Coti Zelati, V. Pata and R. Quintanilla, Regular global attractors of type III thermoelastic extensible beams, Chin. Ann. Math. Series B, 31 (2010), 619-630. doi: 10.1007/s11401-010-0605-4.

[12]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.

[13]

G. Fichera, Is the Fourier theory of heat propagation paradoxical?, Rend. Circ. Mat. Palermo, 41 (1992), 5-28.

[14]

S. Gatti, A. Miranville, V. Pata and S. Zelik, Attractors for semilinear equations of viscoelasticity with very low dissipation, Rocky Mountain J. Math., 38 (2008), 1117-1138. doi: 10.1216/RMJ-2008-38-4-1117.

[15]

S. Gatti, A. Miranville, V. Pata and S. Zelik, Continuous families of exponential attractors for singularly perturbed equations with memory, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 329-366. doi: 10.1017/S0308210509000365.

[16]

C. Giorgi, M. G. Naso, V. Pata and M. Potomkin, Global attractors for the extensible thermoelastic beam system, J. Diff. Eqs, 246 (2009), 3496-3517. doi: 10.1016/j.jde.2009.02.020.

[17]

C. Giorgi, V. Pata and E. Vuk, On the extensible viscoelastic beam, Nonlinearity, 21 (2008), 713-733. doi: 10.1088/0951-7715/21/4/004.

[18]

A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses, 15 (1992), 253-264. doi: 10.1080/01495739208946136.

[19]

A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208. doi: 10.1007/BF00044969.

[20]

A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. I. Classical continuum physics, Proc. Roy. Soc. London A, 448 (1995), 335-356. doi: 10.1098/rspa.1995.0020.

[21]

A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. II. Generalized continua, Proc. Roy. Soc. London A, 448 (1995), 357-377. doi: 10.1098/rspa.1995.0021.

[22]

A. E. Green and P. M. Naghdi, A unified procedure for construction of theories of deformable media. III. Mixtures of interacting continua, Proc. Roy. Soc. London A, 448 (1995), 379-388. doi: 10.1098/rspa.1995.0022.

[23]

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.

[24]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," American Mathematical Society, Providence, 1988.

[25]

A. Haraux, "Syst\`emes Dynamiques Dissipatifs et Applications," Coll. RMA no.17, Masson, Paris, 1991.

[26]

O. A. Ladyzhenskaya, Finding minimal global attractors for the Navier-Stokes equations and other partial differential equations, Russian Math. Surveys, 42 (1987), 27-73. doi: 10.1070/RM1987v042n06ABEH001503.

[27]

V. Pata, Exponential stability in linear viscoelasticity, Quart. Appl. Math., 64 (2006), 499-513.

[28]

V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333-360. doi: 10.1007/s00032-009-0098-3.

[29]

V. Pata, Uniform estimates of Gronwall type, J. Math. Anal. Appl., 373 (2011), 264-270. doi: 10.1016/j.jmaa.2010.07.006.

[30]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.

[31]

M. Potomkin, Asymptotic behavior of thermoviscoelastic Berger plate, Commun. Pure Appl. Anal., 9 (2010), 161-192.

[32]

B. Straughan, "Heat Waves," Springer, New York, 2011.

[33]

R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," Springer, New York, 1997.

[34]

S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36.

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