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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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

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

[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.  Google Scholar

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

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

[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.  Google Scholar

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

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