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Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation
Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models
1. | Department of Applied Mathematics, Donghua University, Shanghai, Songjiang, 201620 |
2. | Department of Mathematics, Nanjing University, Nanjing 210093 |
References:
[1] |
J. Arrieta, A. N. Carvalho and J. K. Hale, A damped hyperbolic equations with critical exponents,, Comm. Partial Differential Equations, 17 (1992), 841.
doi: 10.1080/03605309208820866. |
[2] |
A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", North-Holland, (1992).
|
[3] |
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.
doi: 10.1016/j.na.2009.09.037. |
[4] |
T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems,, Nonlinear Anal., 64 (2006), 484.
doi: 10.1016/j.na.2005.03.111. |
[5] |
T. Caraballo, J. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory,, Discrete Contin. Dyn. Syst., 9 (2008), 525.
doi: 10.3934/dcdsb.2008.9.525. |
[6] |
T. Caraballo, M. J. Garrido-Atienza and B. Schmalfß, Nonautonomous and random attractors for delay random semilinear equations without uniqueness,, Discrete Contin. Dyn. Syst., 21 (2008), 415.
doi: 10.3934/dcds.2008.21.415. |
[7] |
T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems,, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 491.
|
[8] |
T. Caraballo, J. A. Langa and J. C. Robinson, Upper semicontinuity of attractors for small random perturbations of dynamical systems,, Commun. Partial Diff. Eqns., 23 (1998), 1557.
doi: 10.1080/03605309808821394. |
[9] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, On the continuity of pullback attractors for evolution processes,, Nonlinear Anal., 71 (2009), 1812.
doi: 10.1016/j.na.2009.01.016. |
[10] |
A. N. Carvalho and J. A. Langa, Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and unstable manifolds,, J. Differential Equations, 223 (2007), 622.
doi: 10.1016/j.jde.2006.08.009. |
[11] |
I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping,, J. Differential Equatins, 252 (2012), 1229.
doi: 10.1016/j.jde.2011.08.022. |
[12] |
I. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with nonlinear boundary/interior damping,, J. Differential Equations, 233 (2007), 42.
doi: 10.1016/j.jde.2006.09.019. |
[13] |
I. Chueshov and I. Lasiecka, "Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping,", Mem. Amer. Math. Soc., 195 (2008).
|
[14] |
C. M. Elliot and I. N. Kostin, Lower semicontinuity of a nonhyperbolic attractor fro the viscous Cahn-Hilliard equation,, Nonlinearity, 9 (1996), 678.
doi: 10.1088/0951-7715/9/3/005. |
[15] |
F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise,, Stochastics and Stochastics Reports, 59 (1996), 21.
|
[16] |
M. Gobbino, Quasilinear degenerate parabolic equations of Kirchhoff type,, Math. Meth. Appl. Sci., 22 (1999), 375.
doi: 10.1002/(SICI)1099-1476(19990325)22:5<375::AID-MMA26>3.0.CO;2-7. |
[17] |
J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", AMS, (1988).
|
[18] |
J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation,, J. Differential Equations, 73 (1988), 197.
doi: 10.1016/0022-0396(88)90104-0. |
[19] |
J. K. Hale and G. Raugel, Lower semicontinuity of the attractor for a singularly perturbed hyperbolic equation,, J. Dynam. Diff. Eq., 2 (1990), 19.
doi: 10.1007/BF01047769. |
[20] |
L. V. Kapitanski and I. N. Kostin, Attractors of nonlinear evolution equations and their approximations,, Leningrad Math. J., 2 (1991), 97.
|
[21] |
V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation,, J. Differential Equation, 247 (2009), 1120.
doi: 10.1016/j.jde.2009.04.010. |
[22] |
A. Kh. Khanmamedov, Global attractors for wave equation with nonlinear interior damping and critical exponents,, J. Differential Equations, 230 (2006), 702.
doi: 10.1016/j.jde.2006.06.001. |
[23] |
A. Kh. Khanmamedov, Global attractors for von Karman equations with nonlinear dissipation,, J. Math. Anal. Appl., 318 (2006), 92.
doi: 10.1016/j.jmaa.2005.05.031. |
[24] |
G. Kirchhoff, "Vorlesungen über Mechanik,", Teubner, (1883). Google Scholar |
[25] |
P. E. Kloeden, P. Marín-Rubio and J. Real, Pullback attractors for a semilinear heat equation in a non-cylindrical domain,, J. Differential Equations, 244 (2008), 2062.
doi: 10.1016/j.jde.2007.10.031. |
[26] |
P. E. Kloeden, J. Real and C. Y. Sun, Pullback attractors for a semilinear heat equation on time-varying domains,, J. Differential Equations, 246 (2009), 4702.
doi: 10.1016/j.jde.2008.11.017. |
[27] |
P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors,, Proc. Roy. Soc. London A, 463 (2007), 163.
doi: 10.1098/rspa.2006.1753. |
[28] |
S. Kolbasin, Attractors for Kirchhoff's equation with a nonlinear damping coefficient,, Nonlinear Anal., 71 (2009), 2361.
doi: 10.1016/j.na.2009.01.187. |
[29] |
I. N. Kostin, Lower semicontinuity of a non-hyperbolic attractor,, J. London Math. Soc., 52 (1995), 568.
doi: 10.1112/jlms/52.3.568. |
[30] |
P. Marin-Rubio, A. M. Marquez-Duran and J. Real, Pullback attractors for globally modified Navier-Stokes equations with infinite delays,, Discrete Cont. Dyn. Syst., 31 (2011), 779.
doi: 10.3934/dcds.2011.31.779. |
[31] |
T. Matsuyama and R. lkehata, On global solution and energy decay for the wave equation of Kirchhoff type with nonlinear damping term,, J. Math. Anal. Appl., 204 (1996), 729.
doi: 10.1006/jmaa.1996.0464. |
[32] |
M. Nakao, An attractor for a nonlinear dissipative wave equation of Kirchhoff type,, J. Math. Anal. Appl., 353 (2009), 652.
doi: 10.1016/j.jmaa.2008.09.010. |
[33] |
M. Nakao and Z. J. Yang, Global attractors for some qusi-linear wave equations with a strong dissipation,, Adv. Math. Sci. Appl., 17 (2007), 89.
|
[34] |
V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations,, Nonlinearity, 19 (2006), 1495.
doi: 10.1088/0951-7715/19/7/001. |
[35] |
J. C. Robinson, Stability of random attractors under perturbation and approximation,, J. Differential Equations, 186 (2002), 652.
doi: 10.1016/S0022-0396(02)00038-4. |
[36] |
Y. J. Wang, C. K. Zhong and S. F. Zhou, Pullback attractors of nonautonomous dynamical systems,, Discrete Contin. Dyn. Syst., 16 (2006), 587.
doi: 10.3934/dcds.2006.16.705. |
[37] |
Y. H. Wang, On the upper semicontinuity of pullback attractors with applications to plate equations,, Commun. Pure Appl. Anal., 9 (2010), 1653.
doi: 10.3934/cpaa.2010.9.1653. |
[38] |
Y. H. Wang and Y. M. Qin, Upper semicontinuity of pullback attractors for nonclassical diffusion equations,, J. Math. Phys., 51 (2010).
doi: 10.1063/1.3277152. |
[39] |
Y. H. Wang, Pullback attractors for nonautonomous wave equations with critical exponent,, Nonlinear Anal., 68 (2008), 365.
doi: 10.1016/j.na.2006.11.002. |
[40] |
Z. J. Yang and Y. Q. Wang, Global attractor for the Kirchhoff type equation with a strong dissipation,, J. Differential Equations, 249 (2010), 3258. Google Scholar |
[41] |
Z. J. Yang, Longtime behavior of the Kirchhoff type equation with strong damping in $\mathbbR^N$,, J. Differential Equations, 242 (2007), 269.
doi: 10.1016/j.jde.2007.08.004. |
show all references
References:
[1] |
J. Arrieta, A. N. Carvalho and J. K. Hale, A damped hyperbolic equations with critical exponents,, Comm. Partial Differential Equations, 17 (1992), 841.
doi: 10.1080/03605309208820866. |
[2] |
A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", North-Holland, (1992).
|
[3] |
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.
doi: 10.1016/j.na.2009.09.037. |
[4] |
T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems,, Nonlinear Anal., 64 (2006), 484.
doi: 10.1016/j.na.2005.03.111. |
[5] |
T. Caraballo, J. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory,, Discrete Contin. Dyn. Syst., 9 (2008), 525.
doi: 10.3934/dcdsb.2008.9.525. |
[6] |
T. Caraballo, M. J. Garrido-Atienza and B. Schmalfß, Nonautonomous and random attractors for delay random semilinear equations without uniqueness,, Discrete Contin. Dyn. Syst., 21 (2008), 415.
doi: 10.3934/dcds.2008.21.415. |
[7] |
T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems,, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 491.
|
[8] |
T. Caraballo, J. A. Langa and J. C. Robinson, Upper semicontinuity of attractors for small random perturbations of dynamical systems,, Commun. Partial Diff. Eqns., 23 (1998), 1557.
doi: 10.1080/03605309808821394. |
[9] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, On the continuity of pullback attractors for evolution processes,, Nonlinear Anal., 71 (2009), 1812.
doi: 10.1016/j.na.2009.01.016. |
[10] |
A. N. Carvalho and J. A. Langa, Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and unstable manifolds,, J. Differential Equations, 223 (2007), 622.
doi: 10.1016/j.jde.2006.08.009. |
[11] |
I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping,, J. Differential Equatins, 252 (2012), 1229.
doi: 10.1016/j.jde.2011.08.022. |
[12] |
I. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with nonlinear boundary/interior damping,, J. Differential Equations, 233 (2007), 42.
doi: 10.1016/j.jde.2006.09.019. |
[13] |
I. Chueshov and I. Lasiecka, "Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping,", Mem. Amer. Math. Soc., 195 (2008).
|
[14] |
C. M. Elliot and I. N. Kostin, Lower semicontinuity of a nonhyperbolic attractor fro the viscous Cahn-Hilliard equation,, Nonlinearity, 9 (1996), 678.
doi: 10.1088/0951-7715/9/3/005. |
[15] |
F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise,, Stochastics and Stochastics Reports, 59 (1996), 21.
|
[16] |
M. Gobbino, Quasilinear degenerate parabolic equations of Kirchhoff type,, Math. Meth. Appl. Sci., 22 (1999), 375.
doi: 10.1002/(SICI)1099-1476(19990325)22:5<375::AID-MMA26>3.0.CO;2-7. |
[17] |
J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", AMS, (1988).
|
[18] |
J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation,, J. Differential Equations, 73 (1988), 197.
doi: 10.1016/0022-0396(88)90104-0. |
[19] |
J. K. Hale and G. Raugel, Lower semicontinuity of the attractor for a singularly perturbed hyperbolic equation,, J. Dynam. Diff. Eq., 2 (1990), 19.
doi: 10.1007/BF01047769. |
[20] |
L. V. Kapitanski and I. N. Kostin, Attractors of nonlinear evolution equations and their approximations,, Leningrad Math. J., 2 (1991), 97.
|
[21] |
V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation,, J. Differential Equation, 247 (2009), 1120.
doi: 10.1016/j.jde.2009.04.010. |
[22] |
A. Kh. Khanmamedov, Global attractors for wave equation with nonlinear interior damping and critical exponents,, J. Differential Equations, 230 (2006), 702.
doi: 10.1016/j.jde.2006.06.001. |
[23] |
A. Kh. Khanmamedov, Global attractors for von Karman equations with nonlinear dissipation,, J. Math. Anal. Appl., 318 (2006), 92.
doi: 10.1016/j.jmaa.2005.05.031. |
[24] |
G. Kirchhoff, "Vorlesungen über Mechanik,", Teubner, (1883). Google Scholar |
[25] |
P. E. Kloeden, P. Marín-Rubio and J. Real, Pullback attractors for a semilinear heat equation in a non-cylindrical domain,, J. Differential Equations, 244 (2008), 2062.
doi: 10.1016/j.jde.2007.10.031. |
[26] |
P. E. Kloeden, J. Real and C. Y. Sun, Pullback attractors for a semilinear heat equation on time-varying domains,, J. Differential Equations, 246 (2009), 4702.
doi: 10.1016/j.jde.2008.11.017. |
[27] |
P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors,, Proc. Roy. Soc. London A, 463 (2007), 163.
doi: 10.1098/rspa.2006.1753. |
[28] |
S. Kolbasin, Attractors for Kirchhoff's equation with a nonlinear damping coefficient,, Nonlinear Anal., 71 (2009), 2361.
doi: 10.1016/j.na.2009.01.187. |
[29] |
I. N. Kostin, Lower semicontinuity of a non-hyperbolic attractor,, J. London Math. Soc., 52 (1995), 568.
doi: 10.1112/jlms/52.3.568. |
[30] |
P. Marin-Rubio, A. M. Marquez-Duran and J. Real, Pullback attractors for globally modified Navier-Stokes equations with infinite delays,, Discrete Cont. Dyn. Syst., 31 (2011), 779.
doi: 10.3934/dcds.2011.31.779. |
[31] |
T. Matsuyama and R. lkehata, On global solution and energy decay for the wave equation of Kirchhoff type with nonlinear damping term,, J. Math. Anal. Appl., 204 (1996), 729.
doi: 10.1006/jmaa.1996.0464. |
[32] |
M. Nakao, An attractor for a nonlinear dissipative wave equation of Kirchhoff type,, J. Math. Anal. Appl., 353 (2009), 652.
doi: 10.1016/j.jmaa.2008.09.010. |
[33] |
M. Nakao and Z. J. Yang, Global attractors for some qusi-linear wave equations with a strong dissipation,, Adv. Math. Sci. Appl., 17 (2007), 89.
|
[34] |
V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations,, Nonlinearity, 19 (2006), 1495.
doi: 10.1088/0951-7715/19/7/001. |
[35] |
J. C. Robinson, Stability of random attractors under perturbation and approximation,, J. Differential Equations, 186 (2002), 652.
doi: 10.1016/S0022-0396(02)00038-4. |
[36] |
Y. J. Wang, C. K. Zhong and S. F. Zhou, Pullback attractors of nonautonomous dynamical systems,, Discrete Contin. Dyn. Syst., 16 (2006), 587.
doi: 10.3934/dcds.2006.16.705. |
[37] |
Y. H. Wang, On the upper semicontinuity of pullback attractors with applications to plate equations,, Commun. Pure Appl. Anal., 9 (2010), 1653.
doi: 10.3934/cpaa.2010.9.1653. |
[38] |
Y. H. Wang and Y. M. Qin, Upper semicontinuity of pullback attractors for nonclassical diffusion equations,, J. Math. Phys., 51 (2010).
doi: 10.1063/1.3277152. |
[39] |
Y. H. Wang, Pullback attractors for nonautonomous wave equations with critical exponent,, Nonlinear Anal., 68 (2008), 365.
doi: 10.1016/j.na.2006.11.002. |
[40] |
Z. J. Yang and Y. Q. Wang, Global attractor for the Kirchhoff type equation with a strong dissipation,, J. Differential Equations, 249 (2010), 3258. Google Scholar |
[41] |
Z. J. Yang, Longtime behavior of the Kirchhoff type equation with strong damping in $\mathbbR^N$,, J. Differential Equations, 242 (2007), 269.
doi: 10.1016/j.jde.2007.08.004. |
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