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Global weak solutions to Landau-Lifshtiz systems with spin-polarized transport
Uniform attractors for non-autonomous plate equations with $ p $-Laplacian perturbation and critical nonlinearities
1. | College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China |
2. | Departamento de Matemática, Universidade Federal de São Carlos, 13565-905 São Carlos SP, Brazil |
3. | Departamento de Ciências, Campus Regional de Goioerê, Universidade Estadual de Maringá, 87360-000, Goioerê, PR, Brazil |
$ p $ |
$ u_{tt} + \Delta^2 u + a_{\epsilon}(t) u_t - \Delta_p u - \Delta u_t + f(u) = g(x,t), $ |
$ \Omega\subset \mathbb{R}^N $ |
References:
[1] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second edition. Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003. |
[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] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[4] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium publications, 49. American Mathematical Society, Providence, 2002. |
[5] |
I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-Posedness and Long-Time Dynamics, Springer Monographs in Mathematics. Springer, New York, 2010.
doi: 10.1007/978-0-387-87712-9. |
[6] |
I. Chueshov and I. Lasiecka,
Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), 469-512.
doi: 10.1007/s10884-004-4289-x. |
[7] |
I. Chueshov and I. Lasiecka,
Existence, uniqueness of weakly solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models, Discrete Contin. Dyn. Syst., 15 (2006), 777-809.
doi: 10.3934/dcds.2006.15.777. |
[8] |
B. W. Feng, X.-G. Yang and Y. M. Qin,
Uniform attractors for a nonautonomous extensible plate equation with a strong damping, Math. Meth. Appl. Sci., 40 (2017), 3479-3492.
doi: 10.1002/mma.4239. |
[9] |
A. Haraux, Systèmes Dynamiques Dissipatifs et Applications, Recherches en Mathématiques Appliquées, 17. Masson, Paris, 1991. |
[10] |
M. A. Jorge Silva and T. F. Ma,
On a viscoelastic plate equation with history setting and perturbation of $p$-Laplacian type, IMA J. Appl. Math., 78 (2013), 1130-1146.
doi: 10.1093/imamat/hxs011. |
[11] |
M. A. Jorge Silva and T. F. Ma, Long-time dynamics for a class of Kirchhoff models with memory, J. Math. Phys., 54 (2013), 021505, 15 pp.
doi: 10.1063/1.4792606. |
[12] |
A. Kh. Khanmamedov,
Global attractors for von Karman equations with nonlinear interior dissipation, J. Math. Anal. Appl., 318 (2006), 92-101.
doi: 10.1016/j.jmaa.2005.05.031. |
[13] |
J. U. Kim,
A boundary thin obstacle problem for a wave equation, Comm. Partial Differential Equations, 14 (1989), 1011-1026.
doi: 10.1080/03605308908820640. |
[14] |
S. S. Lu,
Attractors for non-autonomous 2D Navier-Stokes equations with less regular normal forces, J. Differential Equations, 230 (2006), 196-212.
doi: 10.1016/j.jde.2006.07.009. |
[15] |
S. S. Lu, H. Q. Wu and C. K. Zhong,
Attractors for non-autonomous 2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst., 13 (2005), 701-719.
doi: 10.3934/dcds.2005.13.701. |
[16] |
S. S. Lu,
Attractors for non-autonomous reactive-diffusion systems with symbols without strong translation compactness, Asymptot. Anal., 54 (2007), 197-210.
|
[17] |
S. Ma, C. K. Zhong and H. T. Song,
Attractors for non-autonomous 2D Navier-Stokes equations with less regular symbols, Nonlinear Anal., 71 (2009), 4215-4222.
doi: 10.1016/j.na.2009.02.107. |
[18] |
S. Ma, X. Y. Cheng and H. T. Li,
Attractors for non-autonomous wave equations with a new class of external forces, J. Math. Anal. Appl., 337 (2008), 808-820.
doi: 10.1016/j.jmaa.2007.03.108. |
[19] |
S. Ma and C. K. Zhong,
The attractors for weakly damped non-autonomous hyperbolic equations with a new class of external forces, Discrete Contin. Dyn. Syst., 18 (2007), 53-70.
doi: 10.3934/dcds.2007.18.53. |
[20] |
T. F. Ma and M. L. Pelicer, Attractors for weakly damped beam equations with $p$-Laplacian, Discrete Contin. Dyn. Syst. Suppl., (2013), 525–534.
doi: 10.3934/proc.2013.2013.525. |
[21] |
C. Y. Sun, D. M. Cao and J. Q. Duan,
Uniform attractors for non-autonomous wave equations with nonlinear damping, SIAM J. Appl. Dyn. Syst., 6 (2007), 293-318.
doi: 10.1137/060663805. |
[22] |
C. Y. Sun, D. M. Cao and J. Q. Duan,
Non-autonomous wave dynamics with memory-asymptotic regularity and uniform attractor, Discrete Contin. Dyn. Syst. B, 9 (2008), 743-761.
doi: 10.3934/dcdsb.2008.9.743. |
[23] |
X. G. Yang, Z. H. Fan and K. Li,
Uniform attractor for non-autonomous Boussinesq-type equation with critical nonlinearity, Math. Meth. Appl. Sci., 39 (2016), 3075-3087.
doi: 10.1002/mma.3753. |
[24] |
Z. J. Yang,
Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term, J. Differential Equations, 187 (2003), 520-540.
doi: 10.1016/S0022-0396(02)00042-6. |
[25] |
Z. J. Yang,
Longtime behavior for a nonlinear wave equation arising in elasto-plastic flow, Math. Meth. Appl. Sci., 32 (2009), 1082-1104.
doi: 10.1002/mma.1080. |
[26] |
Z. J. Yang, Finite-dimensional attractors for the Kirchhoff models with critical exponents, J. Math. Phys., 53 (2012), 032702, 15 pp.
doi: 10.1063/1.3694730. |
[27] |
Z. J. Yang and B. Jin, Global attractor for a class of Kirchhoff models, J. Math. Phys., 50 (2009), 032701, 29 pp. Google Scholar |
[28] |
S. Zelik,
Strong uniform attractors for non-autonomous dissipative PDEs with non translation-compact external force, Discrete Contin. Dyn. Syst., 20 (2015), 781-810.
doi: 10.3934/dcdsb.2015.20.781. |
show all references
References:
[1] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second edition. Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003. |
[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] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[4] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium publications, 49. American Mathematical Society, Providence, 2002. |
[5] |
I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-Posedness and Long-Time Dynamics, Springer Monographs in Mathematics. Springer, New York, 2010.
doi: 10.1007/978-0-387-87712-9. |
[6] |
I. Chueshov and I. Lasiecka,
Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), 469-512.
doi: 10.1007/s10884-004-4289-x. |
[7] |
I. Chueshov and I. Lasiecka,
Existence, uniqueness of weakly solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models, Discrete Contin. Dyn. Syst., 15 (2006), 777-809.
doi: 10.3934/dcds.2006.15.777. |
[8] |
B. W. Feng, X.-G. Yang and Y. M. Qin,
Uniform attractors for a nonautonomous extensible plate equation with a strong damping, Math. Meth. Appl. Sci., 40 (2017), 3479-3492.
doi: 10.1002/mma.4239. |
[9] |
A. Haraux, Systèmes Dynamiques Dissipatifs et Applications, Recherches en Mathématiques Appliquées, 17. Masson, Paris, 1991. |
[10] |
M. A. Jorge Silva and T. F. Ma,
On a viscoelastic plate equation with history setting and perturbation of $p$-Laplacian type, IMA J. Appl. Math., 78 (2013), 1130-1146.
doi: 10.1093/imamat/hxs011. |
[11] |
M. A. Jorge Silva and T. F. Ma, Long-time dynamics for a class of Kirchhoff models with memory, J. Math. Phys., 54 (2013), 021505, 15 pp.
doi: 10.1063/1.4792606. |
[12] |
A. Kh. Khanmamedov,
Global attractors for von Karman equations with nonlinear interior dissipation, J. Math. Anal. Appl., 318 (2006), 92-101.
doi: 10.1016/j.jmaa.2005.05.031. |
[13] |
J. U. Kim,
A boundary thin obstacle problem for a wave equation, Comm. Partial Differential Equations, 14 (1989), 1011-1026.
doi: 10.1080/03605308908820640. |
[14] |
S. S. Lu,
Attractors for non-autonomous 2D Navier-Stokes equations with less regular normal forces, J. Differential Equations, 230 (2006), 196-212.
doi: 10.1016/j.jde.2006.07.009. |
[15] |
S. S. Lu, H. Q. Wu and C. K. Zhong,
Attractors for non-autonomous 2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst., 13 (2005), 701-719.
doi: 10.3934/dcds.2005.13.701. |
[16] |
S. S. Lu,
Attractors for non-autonomous reactive-diffusion systems with symbols without strong translation compactness, Asymptot. Anal., 54 (2007), 197-210.
|
[17] |
S. Ma, C. K. Zhong and H. T. Song,
Attractors for non-autonomous 2D Navier-Stokes equations with less regular symbols, Nonlinear Anal., 71 (2009), 4215-4222.
doi: 10.1016/j.na.2009.02.107. |
[18] |
S. Ma, X. Y. Cheng and H. T. Li,
Attractors for non-autonomous wave equations with a new class of external forces, J. Math. Anal. Appl., 337 (2008), 808-820.
doi: 10.1016/j.jmaa.2007.03.108. |
[19] |
S. Ma and C. K. Zhong,
The attractors for weakly damped non-autonomous hyperbolic equations with a new class of external forces, Discrete Contin. Dyn. Syst., 18 (2007), 53-70.
doi: 10.3934/dcds.2007.18.53. |
[20] |
T. F. Ma and M. L. Pelicer, Attractors for weakly damped beam equations with $p$-Laplacian, Discrete Contin. Dyn. Syst. Suppl., (2013), 525–534.
doi: 10.3934/proc.2013.2013.525. |
[21] |
C. Y. Sun, D. M. Cao and J. Q. Duan,
Uniform attractors for non-autonomous wave equations with nonlinear damping, SIAM J. Appl. Dyn. Syst., 6 (2007), 293-318.
doi: 10.1137/060663805. |
[22] |
C. Y. Sun, D. M. Cao and J. Q. Duan,
Non-autonomous wave dynamics with memory-asymptotic regularity and uniform attractor, Discrete Contin. Dyn. Syst. B, 9 (2008), 743-761.
doi: 10.3934/dcdsb.2008.9.743. |
[23] |
X. G. Yang, Z. H. Fan and K. Li,
Uniform attractor for non-autonomous Boussinesq-type equation with critical nonlinearity, Math. Meth. Appl. Sci., 39 (2016), 3075-3087.
doi: 10.1002/mma.3753. |
[24] |
Z. J. Yang,
Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term, J. Differential Equations, 187 (2003), 520-540.
doi: 10.1016/S0022-0396(02)00042-6. |
[25] |
Z. J. Yang,
Longtime behavior for a nonlinear wave equation arising in elasto-plastic flow, Math. Meth. Appl. Sci., 32 (2009), 1082-1104.
doi: 10.1002/mma.1080. |
[26] |
Z. J. Yang, Finite-dimensional attractors for the Kirchhoff models with critical exponents, J. Math. Phys., 53 (2012), 032702, 15 pp.
doi: 10.1063/1.3694730. |
[27] |
Z. J. Yang and B. Jin, Global attractor for a class of Kirchhoff models, J. Math. Phys., 50 (2009), 032701, 29 pp. Google Scholar |
[28] |
S. Zelik,
Strong uniform attractors for non-autonomous dissipative PDEs with non translation-compact external force, Discrete Contin. Dyn. Syst., 20 (2015), 781-810.
doi: 10.3934/dcdsb.2015.20.781. |
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