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Pullback attractors for non-autonomous evolution equations with spatially variable exponents
1. | School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China |
2. | Instituto de Matemática e Computaçã, Universidade Federal de Itajubá, 37500-903 Itajubá MG, Brazil |
References:
[1] |
R. Aboulaich, D. Meskine, A. Souissi, New diffusion models in image processing, Computers and Mathematics with Applications, 56 (2008), 874-882.
doi: 10.1016/j.camwa.2008.01.017. |
[2] |
G. Akagi and K. Matsuura, Well-posedness and large-time behaviors of solutions for a parabolic equation involving $p(x)-$Laplacian, Discrete Contin. Dyn. Syst. 2011, Dynamical systems, differential equations and applications. 8th AIMS Conference, Suppl. Vol. I (2011), 22-31. |
[3] |
G. Akagi and K. Matsuura, Nonlinear diffusion equations driven by $p(\cdot)-$Laplacian, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 37-64.
doi: 10.1007/s00030-012-0153-6. |
[4] |
B. Amaziane, L. Pankratov and A. Piatnitski, Nonlinear flow through double porosity media in variable exponent Sobolev spaces, Nonlinear Anal.: Real World Applications, 10 (2009), 2521-2530.
doi: 10.1016/j.nonrwa.2008.05.008. |
[5] |
S. N. Antonstev and S. I. Shmarev, A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions, Nonlinear Analysis, 60 (2005), 515-545.
doi: 10.1016/j.na.2004.09.026. |
[6] |
S. N. Antontsev and S. Shmarev, Anisotropic parabolic equations with variable nonlinearity, Publications Mathematiques, 53 (2009), 355-399.
doi: 10.5565/PUBLMAT-53209-04. |
[7] |
S. N. Antontsev and S. Shmarev, Existence and uniqueness of solutions of degenerate parabolic equations with variable exponents of nonlinearity, J. Mathematical Sciences, 150 (2008), 2289-2301.
doi: 10.1007/s10958-008-0129-6. |
[8] |
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Space, Noordhoff International Publishing, 1976. |
[9] |
H. Brèzis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Amsterdam: North-Holland Publishing Company, 1973. |
[10] |
E. Capelato, K. Schiabel-Silva and R. P. Silva, Perturbation of a nonautonomous problem in $\mathbbR^n$, Math. Meth. Appl. Sci., 36 (2013), 1625-1630.
doi: 10.1002/mma.2712. |
[11] |
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. |
[12] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences 182, Springer-Verlag, 2012.
doi: 10.1007/978-1-4614-4581-4. |
[13] |
G. Chen and C. Zhong, Uniform attractors for non-autonomous $p-$Laplacian equations, Nonlinear Anal., 68 (2008), 3349-3363.
doi: 10.1016/j.na.2007.03.025. |
[14] |
Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Math., 66 (2006), 1383-1406.
doi: 10.1137/050624522. |
[15] |
L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, Berlin, Heidelberg, 2011.
doi: 10.1007/978-3-642-18363-8. |
[16] |
M. Effendiev, Attractors for Degenerate Parabolic type Equations, Mathematical Surveys and Monographs, 192. American math. Soc., Providence, RI, 2013. |
[17] |
F. Ettwein and M. Růžička, Existence of local strong solutions for motions of electrorheological fluids in three dimensions, Computers and mathematics with applications, 53 (2007), 595-604.
doi: 10.1016/j.camwa.2006.02.032. |
[18] |
X. L. Fan and Q. H. Zhang, Existence of solutions for $p(x)-$laplacian Dirichlet problems, Nonlinear Anal., 52 (2003), 1843-1852.
doi: 10.1016/S0362-546X(02)00150-5. |
[19] |
X. L. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega)$, J. Math. Anal. Appl., 262 (2001), 749-760.
doi: 10.1006/jmaa.2001.7618. |
[20] |
X. L. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2001), 424-446.
doi: 10.1006/jmaa.2000.7617. |
[21] |
Z. Guo, Q. Liu, J. Sun and B. Wu, Reaction-diffusion systems with $p(x)-$growth for image denoising, Nonlinear Analysis: Real World Applications, 12 (2011), 2904-2918.
doi: 10.1016/j.nonrwa.2011.04.015. |
[22] |
P. Harjulehto, P. Hästö, U. Lê and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Analysis, 72 (2010), 4551-4574.
doi: 10.1016/j.na.2010.02.033. |
[23] |
P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176. Amer. Math. Soc., Providence, 2011. |
[24] |
S. Lian, W. Gao, H. Yuan and C. Cao, Existence of solutions to an initial Dirichlet problem of evolutional $p(x)-$Laplace equations, Ann. I. H. Poincaré - AN, 29 (2012), 377-399.
doi: 10.1016/j.anihpc.2012.01.001. |
[25] |
Y. Li and C. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations, Appl. Math. Comput., 190 (2007), 1020-1029.
doi: 10.1016/j.amc.2006.11.187. |
[26] |
W. Niu, Long-time behavior for a nonlinear parabolic problem with variable exponents, J. Math. Anal. Appl., 393 (2012), 56-65.
doi: 10.1016/j.jmaa.2012.03.039. |
[27] |
K. Rajagopal and M. Růžička, Mathematical modelling of electrorheological materials, Contin. Mech. Thermodyn., 13 (2001), 59-78.
doi: 10.1007/s001610100034. |
[28] |
M. Růžička, Flow of shear dependent electrorheological fluids, C. R. Acad. Sci. Paris Sér. I, 329 (1999), 393-398.
doi: 10.1016/S0764-4442(00)88612-7. |
[29] |
M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, in: Lectures Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000.
doi: 10.1007/BFb0104029. |
[30] |
J. Simsen, A global attractor for a $p(x)$-Laplacian problem, Nonlinear Anal., 73 (2010), 3278-3283.
doi: 10.1016/j.na.2010.06.087. |
[31] |
J. Simsen and C. B. Gentile, Well-posed $p$-laplacian problems with large diffusion, Nonlinear Anal., 71 (2009), 4609-4617.
doi: 10.1016/j.na.2009.03.041. |
[32] |
J. Simsen and M. S. Simsen, PDE and ODE Limit Problems for $p(x)$-Laplacian Parabolic Equations, J. Math. Anal. Appl., 383 (2011), 71-81.
doi: 10.1016/j.jmaa.2011.05.003. |
[33] |
J. Simsen, M. S. Simsen and M. R. T. Primo, Continuity of the flows and upper semicontinuity of global attractors for $p_s(x)$-Laplacian parabolic problems, J. Math. Anal. Appl., 398 (2013), 138-150.
doi: 10.1016/j.jmaa.2012.08.047. |
[34] |
J. Simsen, M. J. D. Nascimento and M. S. Simsen, Existence and upper semicontinuity of pullback attractors for non-autonomous p-Laplacian parabolic problems, J. Math. Anal. Appl., 413 (2014), 685-699.
doi: 10.1016/j.jmaa.2013.12.019. |
[35] |
L. Songzhe, G. Wenjie, C. Chunling and Y. Hongjun, Study of the solutions to a model porous medium equation with variable exponent of nonlinearity, J. Math. Anal. Appl., 342 (2008), 27-38.
doi: 10.1016/j.jmaa.2007.11.046. |
[36] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[37] |
B. Wang and R. Jones, Asymptotic behavior of a class of non-autonomous degenerate parabolic equations, Nonlinear Anal., 72 (2010), 3887-3902.
doi: 10.1016/j.na.2010.01.026. |
[38] |
L. Yang, M. Yang and P. E. Kloeden, Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions, Discrete Contin. Dyn. Syst. Series B, 17 (2012), 2635-2651.
doi: 10.3934/dcdsb.2012.17.2635. |
[39] |
S. Yotsutani, Evolution equations associated with the subdifferentials, J. Math. Soc. Japan, 31 (1978), 623-646.
doi: 10.2969/jmsj/03140623. |
show all references
References:
[1] |
R. Aboulaich, D. Meskine, A. Souissi, New diffusion models in image processing, Computers and Mathematics with Applications, 56 (2008), 874-882.
doi: 10.1016/j.camwa.2008.01.017. |
[2] |
G. Akagi and K. Matsuura, Well-posedness and large-time behaviors of solutions for a parabolic equation involving $p(x)-$Laplacian, Discrete Contin. Dyn. Syst. 2011, Dynamical systems, differential equations and applications. 8th AIMS Conference, Suppl. Vol. I (2011), 22-31. |
[3] |
G. Akagi and K. Matsuura, Nonlinear diffusion equations driven by $p(\cdot)-$Laplacian, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 37-64.
doi: 10.1007/s00030-012-0153-6. |
[4] |
B. Amaziane, L. Pankratov and A. Piatnitski, Nonlinear flow through double porosity media in variable exponent Sobolev spaces, Nonlinear Anal.: Real World Applications, 10 (2009), 2521-2530.
doi: 10.1016/j.nonrwa.2008.05.008. |
[5] |
S. N. Antonstev and S. I. Shmarev, A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions, Nonlinear Analysis, 60 (2005), 515-545.
doi: 10.1016/j.na.2004.09.026. |
[6] |
S. N. Antontsev and S. Shmarev, Anisotropic parabolic equations with variable nonlinearity, Publications Mathematiques, 53 (2009), 355-399.
doi: 10.5565/PUBLMAT-53209-04. |
[7] |
S. N. Antontsev and S. Shmarev, Existence and uniqueness of solutions of degenerate parabolic equations with variable exponents of nonlinearity, J. Mathematical Sciences, 150 (2008), 2289-2301.
doi: 10.1007/s10958-008-0129-6. |
[8] |
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Space, Noordhoff International Publishing, 1976. |
[9] |
H. Brèzis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Amsterdam: North-Holland Publishing Company, 1973. |
[10] |
E. Capelato, K. Schiabel-Silva and R. P. Silva, Perturbation of a nonautonomous problem in $\mathbbR^n$, Math. Meth. Appl. Sci., 36 (2013), 1625-1630.
doi: 10.1002/mma.2712. |
[11] |
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. |
[12] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences 182, Springer-Verlag, 2012.
doi: 10.1007/978-1-4614-4581-4. |
[13] |
G. Chen and C. Zhong, Uniform attractors for non-autonomous $p-$Laplacian equations, Nonlinear Anal., 68 (2008), 3349-3363.
doi: 10.1016/j.na.2007.03.025. |
[14] |
Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Math., 66 (2006), 1383-1406.
doi: 10.1137/050624522. |
[15] |
L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, Berlin, Heidelberg, 2011.
doi: 10.1007/978-3-642-18363-8. |
[16] |
M. Effendiev, Attractors for Degenerate Parabolic type Equations, Mathematical Surveys and Monographs, 192. American math. Soc., Providence, RI, 2013. |
[17] |
F. Ettwein and M. Růžička, Existence of local strong solutions for motions of electrorheological fluids in three dimensions, Computers and mathematics with applications, 53 (2007), 595-604.
doi: 10.1016/j.camwa.2006.02.032. |
[18] |
X. L. Fan and Q. H. Zhang, Existence of solutions for $p(x)-$laplacian Dirichlet problems, Nonlinear Anal., 52 (2003), 1843-1852.
doi: 10.1016/S0362-546X(02)00150-5. |
[19] |
X. L. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega)$, J. Math. Anal. Appl., 262 (2001), 749-760.
doi: 10.1006/jmaa.2001.7618. |
[20] |
X. L. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2001), 424-446.
doi: 10.1006/jmaa.2000.7617. |
[21] |
Z. Guo, Q. Liu, J. Sun and B. Wu, Reaction-diffusion systems with $p(x)-$growth for image denoising, Nonlinear Analysis: Real World Applications, 12 (2011), 2904-2918.
doi: 10.1016/j.nonrwa.2011.04.015. |
[22] |
P. Harjulehto, P. Hästö, U. Lê and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Analysis, 72 (2010), 4551-4574.
doi: 10.1016/j.na.2010.02.033. |
[23] |
P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176. Amer. Math. Soc., Providence, 2011. |
[24] |
S. Lian, W. Gao, H. Yuan and C. Cao, Existence of solutions to an initial Dirichlet problem of evolutional $p(x)-$Laplace equations, Ann. I. H. Poincaré - AN, 29 (2012), 377-399.
doi: 10.1016/j.anihpc.2012.01.001. |
[25] |
Y. Li and C. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations, Appl. Math. Comput., 190 (2007), 1020-1029.
doi: 10.1016/j.amc.2006.11.187. |
[26] |
W. Niu, Long-time behavior for a nonlinear parabolic problem with variable exponents, J. Math. Anal. Appl., 393 (2012), 56-65.
doi: 10.1016/j.jmaa.2012.03.039. |
[27] |
K. Rajagopal and M. Růžička, Mathematical modelling of electrorheological materials, Contin. Mech. Thermodyn., 13 (2001), 59-78.
doi: 10.1007/s001610100034. |
[28] |
M. Růžička, Flow of shear dependent electrorheological fluids, C. R. Acad. Sci. Paris Sér. I, 329 (1999), 393-398.
doi: 10.1016/S0764-4442(00)88612-7. |
[29] |
M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, in: Lectures Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000.
doi: 10.1007/BFb0104029. |
[30] |
J. Simsen, A global attractor for a $p(x)$-Laplacian problem, Nonlinear Anal., 73 (2010), 3278-3283.
doi: 10.1016/j.na.2010.06.087. |
[31] |
J. Simsen and C. B. Gentile, Well-posed $p$-laplacian problems with large diffusion, Nonlinear Anal., 71 (2009), 4609-4617.
doi: 10.1016/j.na.2009.03.041. |
[32] |
J. Simsen and M. S. Simsen, PDE and ODE Limit Problems for $p(x)$-Laplacian Parabolic Equations, J. Math. Anal. Appl., 383 (2011), 71-81.
doi: 10.1016/j.jmaa.2011.05.003. |
[33] |
J. Simsen, M. S. Simsen and M. R. T. Primo, Continuity of the flows and upper semicontinuity of global attractors for $p_s(x)$-Laplacian parabolic problems, J. Math. Anal. Appl., 398 (2013), 138-150.
doi: 10.1016/j.jmaa.2012.08.047. |
[34] |
J. Simsen, M. J. D. Nascimento and M. S. Simsen, Existence and upper semicontinuity of pullback attractors for non-autonomous p-Laplacian parabolic problems, J. Math. Anal. Appl., 413 (2014), 685-699.
doi: 10.1016/j.jmaa.2013.12.019. |
[35] |
L. Songzhe, G. Wenjie, C. Chunling and Y. Hongjun, Study of the solutions to a model porous medium equation with variable exponent of nonlinearity, J. Math. Anal. Appl., 342 (2008), 27-38.
doi: 10.1016/j.jmaa.2007.11.046. |
[36] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[37] |
B. Wang and R. Jones, Asymptotic behavior of a class of non-autonomous degenerate parabolic equations, Nonlinear Anal., 72 (2010), 3887-3902.
doi: 10.1016/j.na.2010.01.026. |
[38] |
L. Yang, M. Yang and P. E. Kloeden, Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions, Discrete Contin. Dyn. Syst. Series B, 17 (2012), 2635-2651.
doi: 10.3934/dcdsb.2012.17.2635. |
[39] |
S. Yotsutani, Evolution equations associated with the subdifferentials, J. Math. Soc. Japan, 31 (1978), 623-646.
doi: 10.2969/jmsj/03140623. |
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