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November  2014, 13(6): 2543-2557. doi: 10.3934/cpaa.2014.13.2543

## 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

Received  February 2014 Revised  June 2014 Published  July 2014

Dissipative problems in electrorheological fluids, porous media and image processing often involve spatially dependent exponents. They also include time-dependent terms as in equation \begin{eqnarray} \frac{\partial u_\lambda}{\partial t}(t)-\textrm{div}(D_\lambda(t)|\nabla u_\lambda(t)|^{p(x)-2}\nabla u_\lambda(t))+|u_\lambda(t)|^{p(x)-2}u_\lambda(t) = B(t,u_\lambda(t)) \end{eqnarray} on a bounded smooth domain $\Omega$ in $\mathbb{R}^n$, $n\geq 1$, with a homogeneous Neumann boundary condition, where the exponent $p(\cdot)\in C(\bar{\Omega}, \mathbb{R}^+)$ satisfying $p^-$ $:=$ $\min p(x)$ $>$ $2$, and $\lambda$ $\in$ $[0,\infty)$ is a parameter.
The existence and upper semicontinuity of pullback attractors are established for this equation under the assumptions, amongst others, that $B$ is globally Lipschitz in its second variable and $D_\lambda$ $\in$ $L^\infty([\tau,T] \times \Omega, \mathbb{R}^+)$ is bounded from above and below, is monotonically nonincreasing in time and continuous in the parameter $\lambda$. The global existence and uniqueness of strong solutions is obtained through results of Yotsutani.
Citation: Peter E. Kloeden, Jacson Simsen. Pullback attractors for non-autonomous evolution equations with spatially variable exponents. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2543-2557. doi: 10.3934/cpaa.2014.13.2543
##### 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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