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November  2018, 23(9): 3553-3571. doi: 10.3934/dcdsb.2017214

Pullback attractors for a class of non-autonomous thermoelastic plate systems

Flank D. M. Bezerra 1, , Vera L. Carbone 2, , Marcelo J. D. Nascimento 2, and  Karina Schiabel 2,,

1. 

Universidade Federal da Paraíba Departamento de Matemática 58051-900 João Pessoa PB, Brazil
2. 

Universidade Federal de São Carlos, Departamento de Matemática, 13565-905 São Carlos SP, Brazil

* Corresponding author

The first author is partially supported by FAPESP grant #2014/03686-3, Brazil.
The third author is partially supported by FAPESP grant #2014/03109-6, Brazil.

Received  April 2017 Revised  July 2017 Published  November 2018 Early access  September 2017

In this article we study the asymptotic behavior of solutions, in the sense of pullback attractors, of the evolution system
$\begin{cases}u_{tt} +Δ^2 u+a(t)Δθ=f(t,u),&t>τ,\ x∈Ω,\\θ_t-κΔ θ-a(t)Δ u_t=0,&t>τ,\ x∈Ω,\end{cases}$
subject to boundary conditions
$u=Δ u=θ=0,\ t>τ,\ x∈\partial Ω,$
where $Ω$ is a bounded domain in $\mathbb{R}^N$ with $N≥ 2$, which boundary $\partialΩ$ is assumed to be a $\mathcal{C}^4$-hypersurface, $κ>0$ is constant, $a$ is an Hölder continuous function and $f$ is a dissipative nonlinearity locally Lipschitz in the second variable. Using the theory of uniform sectorial operators, in the sense of P. Sobolevskiǐ ([23]), we give a partial description of the fractional power spaces scale for the thermoelastic plate operator and we show the local and global well-posedness of this non-autonomous problem. Furthermore we prove existence and uniform boundedness of pullback attractors.
Keywords: Pullback attractors, local existence, global existence, thermoelastic plate, non-autonomous system.
Mathematics Subject Classification: Primary: 37B55, 34D45; Secondary: 35B40, 35B41.
Citation: Flank D. M. Bezerra, Vera L. Carbone, Marcelo J. D. Nascimento, Karina Schiabel. Pullback attractors for a class of non-autonomous thermoelastic plate systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3553-3571. doi: 10.3934/dcdsb.2017214
References:
[1]

H. Amann, Linear and Quasilinear Parabolic Problems. Volume Ⅰ: Abstract Linear Theory Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9221-6.

[2]

D. Andrade, M. A. Jorge Silva and T. F. Ma, Exponential stability for a plate equation with p-Laplacian and memory terms, Math. Meth. Appl. Sci., 35 (2012), 417-426.  doi: 10.1002/mma.1552.

[3]

R. O. Araújo, To Fu Ma and Y. Qin, Long-time behavior of a quasilinear viscoelastic equation with past history, J. Differential Equations, 254 (2013), 4066-4087.  doi: 10.1016/j.jde.2013.02.010.

[4]

A. R. A. Barbosa and T. F. 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]

M. Baroun, S. Boulite, T. Diagana and L. Maniar, Almost periodic solutions to some semilinear non- autonomous thermoelastic plate equations, J. Math. Anal. Appl., 349 (2009), 74-84.  doi: 10.1016/j.jmaa.2008.08.034.

[6]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. Ferreira, Existence and uniform decay for a non-linear viscoelastic equation with strong damping, Math. Methods Appl. Sci., 24 (2001), 1043-1053.  doi: 10.1002/mma.250.

[7]

T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor, Nonlinear Anal., 74 (2011), 2272-2283.  doi: 10.1016/j.na.2010.11.032.

[8]

T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Anal., 72 (2010), 1967-1976.  doi: 10.1016/j.na.2009.09.037.

[9]

T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis, 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.

[10]

V. L. Carbone, M. J. D. Nascimento, K. Schiabel-Silva and R. P. Silva, Pullback attractors for a singularly nonautonomous plate equation, Electron. J. Differential Equations, 77 (2011), 1-13. 

[11]

A. N. Carvalho and J. W. Cholewa, Local well-posedness for strongly damped wave equations with critical nonlinearities, Bull. Austral. Math. Soc., 66 (2002), 443-463.  doi: 10.1017/S0004972700040296.

[12]

A. N. Carvalho, J. W. Cholewa and T. Dlotko, Damped wave equations with fast growing dissipative nonlinearities, Discrete Contin. Dyn. Syst., 24 (2009), 1147-1165.  doi: 10.3934/dcds.2009.24.1147.

[13]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems Applied Mathematical Sciences, 182, Springer-Verlag, 2013. doi: 10.1007/978-1-4614-4581-4.

[14]

A. N. Carvalho and M. J. D. Nascimento, Singularly non-autonomous semilinear parabolic problems with critical exponents and applications, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 449-471. 

[15]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics AMS Colloquium Publications v. 49, A. M. S, Providence, 2002.

[16]

C. Giorgi, J. E. Muñoz Rivera and V. Pata, Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl., 260 (2001), 83-99.  doi: 10.1006/jmaa.2001.7437.

[17]

D. Henry, Geometric Theory of Semilinear Parabolic Equations Lecture Notes in Mathematics 840, Springer-Verlag, Berlin, 1981.

[18]

I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with free boundary conditions, Ann. Sc. Norm. Super. Pisa Cl. Sci., 27 (1998), 457-482. 

[19]

I. Lasiecka and R. Triggiani, Analyticity, and lack thereof, on thermo-elastic semigroups, Contrȏle et Équations aux Dérivées Partielles, ESAIM: Proceedings, 4 (1998), 199-222. 

[20]

Z. Y. Liu and M. Renardy, A note on the equations of a thermoelastic plate, Appl. Math. Lett., 8 (1995), 1-6.  doi: 10.1016/0893-9659(95)00020-Q.

[21]

Z. Liu and S. Zheng, xponential stability of the Kirchhoff plate with thermal or viscoelastic damping, Quart. Appl. Math., 55 (1997), 551-564.  doi: 10.1090/qam/1466148.

[22]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems In CRC Research Notes in Mathematics 398, Chapman and Hall, 1999.

[23]

P. E. Sobolevskiǐ, Equations of parabolic type in a Banach space, Amer. Math. Soc. Transl., 49 (1966), 1-62. 

[24]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators Veb Deutscher, 1978.

show all references

References:
[1]

H. Amann, Linear and Quasilinear Parabolic Problems. Volume Ⅰ: Abstract Linear Theory Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9221-6.

[2]

D. Andrade, M. A. Jorge Silva and T. F. Ma, Exponential stability for a plate equation with p-Laplacian and memory terms, Math. Meth. Appl. Sci., 35 (2012), 417-426.  doi: 10.1002/mma.1552.

[3]

R. O. Araújo, To Fu Ma and Y. Qin, Long-time behavior of a quasilinear viscoelastic equation with past history, J. Differential Equations, 254 (2013), 4066-4087.  doi: 10.1016/j.jde.2013.02.010.

[4]

A. R. A. Barbosa and T. F. 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]

M. Baroun, S. Boulite, T. Diagana and L. Maniar, Almost periodic solutions to some semilinear non- autonomous thermoelastic plate equations, J. Math. Anal. Appl., 349 (2009), 74-84.  doi: 10.1016/j.jmaa.2008.08.034.

[6]

M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. Ferreira, Existence and uniform decay for a non-linear viscoelastic equation with strong damping, Math. Methods Appl. Sci., 24 (2001), 1043-1053.  doi: 10.1002/mma.250.

[7]

T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor, Nonlinear Anal., 74 (2011), 2272-2283.  doi: 10.1016/j.na.2010.11.032.

[8]

T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Anal., 72 (2010), 1967-1976.  doi: 10.1016/j.na.2009.09.037.

[9]

T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis, 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.

[10]

V. L. Carbone, M. J. D. Nascimento, K. Schiabel-Silva and R. P. Silva, Pullback attractors for a singularly nonautonomous plate equation, Electron. J. Differential Equations, 77 (2011), 1-13. 

[11]

A. N. Carvalho and J. W. Cholewa, Local well-posedness for strongly damped wave equations with critical nonlinearities, Bull. Austral. Math. Soc., 66 (2002), 443-463.  doi: 10.1017/S0004972700040296.

[12]

A. N. Carvalho, J. W. Cholewa and T. Dlotko, Damped wave equations with fast growing dissipative nonlinearities, Discrete Contin. Dyn. Syst., 24 (2009), 1147-1165.  doi: 10.3934/dcds.2009.24.1147.

[13]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems Applied Mathematical Sciences, 182, Springer-Verlag, 2013. doi: 10.1007/978-1-4614-4581-4.

[14]

A. N. Carvalho and M. J. D. Nascimento, Singularly non-autonomous semilinear parabolic problems with critical exponents and applications, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 449-471. 

[15]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics AMS Colloquium Publications v. 49, A. M. S, Providence, 2002.

[16]

C. Giorgi, J. E. Muñoz Rivera and V. Pata, Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl., 260 (2001), 83-99.  doi: 10.1006/jmaa.2001.7437.

[17]

D. Henry, Geometric Theory of Semilinear Parabolic Equations Lecture Notes in Mathematics 840, Springer-Verlag, Berlin, 1981.

[18]

I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with free boundary conditions, Ann. Sc. Norm. Super. Pisa Cl. Sci., 27 (1998), 457-482. 

[19]

I. Lasiecka and R. Triggiani, Analyticity, and lack thereof, on thermo-elastic semigroups, Contrȏle et Équations aux Dérivées Partielles, ESAIM: Proceedings, 4 (1998), 199-222. 

[20]

Z. Y. Liu and M. Renardy, A note on the equations of a thermoelastic plate, Appl. Math. Lett., 8 (1995), 1-6.  doi: 10.1016/0893-9659(95)00020-Q.

[21]

Z. Liu and S. Zheng, xponential stability of the Kirchhoff plate with thermal or viscoelastic damping, Quart. Appl. Math., 55 (1997), 551-564.  doi: 10.1090/qam/1466148.

[22]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems In CRC Research Notes in Mathematics 398, Chapman and Hall, 1999.

[23]

P. E. Sobolevskiǐ, Equations of parabolic type in a Banach space, Amer. Math. Soc. Transl., 49 (1966), 1-62. 

[24]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators Veb Deutscher, 1978.

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