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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 |
$\begin{cases}u_{tt} +Δ^2 u+a(t)Δθ=f(t,u),&t>τ,\ x∈Ω,\\θ_t-κΔ θ-a(t)Δ u_t=0,&t>τ,\ x∈Ω,\end{cases}$ |
$u=Δ u=θ=0,\ t>τ,\ x∈\partial Ω,$ |
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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