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

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

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.
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 & 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.  Google Scholar

[2]

D. AndradeM. 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.  Google Scholar

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R. O. AraújoTo 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.  Google Scholar

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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.  Google Scholar

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M. BarounS. BouliteT. 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.  Google Scholar

[6]

M. M. CavalcantiV. 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.  Google Scholar

[7]

T. CaraballoA. N. CarvalhoJ. 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.  Google Scholar

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T. CaraballoA. N. CarvalhoJ. 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.  Google Scholar

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T. CaraballoG. 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.  Google Scholar

[10]

V. L. CarboneM. J. D. NascimentoK. Schiabel-Silva and R. P. Silva, Pullback attractors for a singularly nonautonomous plate equation, Electron. J. Differential Equations, 77 (2011), 1-13.   Google Scholar

[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.  Google Scholar

[12]

A. N. CarvalhoJ. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[15]

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

[16]

C. GiorgiJ. 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.  Google Scholar

[17]

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

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[24]

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

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.  Google Scholar

[2]

D. AndradeM. 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.  Google Scholar

[3]

R. O. AraújoTo 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.  Google Scholar

[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.  Google Scholar

[5]

M. BarounS. BouliteT. 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.  Google Scholar

[6]

M. M. CavalcantiV. 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.  Google Scholar

[7]

T. CaraballoA. N. CarvalhoJ. 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.  Google Scholar

[8]

T. CaraballoA. N. CarvalhoJ. 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.  Google Scholar

[9]

T. CaraballoG. 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.  Google Scholar

[10]

V. L. CarboneM. J. D. NascimentoK. Schiabel-Silva and R. P. Silva, Pullback attractors for a singularly nonautonomous plate equation, Electron. J. Differential Equations, 77 (2011), 1-13.   Google Scholar

[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.  Google Scholar

[12]

A. N. CarvalhoJ. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[15]

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

[16]

C. GiorgiJ. 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.  Google Scholar

[17]

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

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[24]

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

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