Pullback attractors for non-autonomous evolution equations with spatially variable exponents

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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

Peter E. Kloeden ^1, and Jacson Simsen ^2,

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

Abstract
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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.

Keywords: upper semicontinuity., variable exponents, pullback attractors, Non-autonomous parabolic problems.

Mathematics Subject Classification: Primary: 35K55, 35K92; Secondary: 35A16, 35B40, 35B41, 37B5.

Citation: Peter E. Kloeden, Jacson Simsen. Pullback attractors for non-autonomous evolution equations with spatially variable exponents. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2543-2557. doi: 10.3934/cpaa.2014.13.2543

References:

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[14]	Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration,, \emph{SIAM J. Math.}, 66 (2006), 1383. doi: 10.1137/050624522.
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[21]	Z. Guo, Q. Liu, J. Sun and B. Wu, Reaction-diffusion systems with $p(x)-$growth for image denoising,, \emph{Nonlinear Analysis: Real World Applications}, 12 (2011), 2904. 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,, \emph{Nonlinear Analysis}, 72 (2010), 4551. doi: 10.1016/j.na.2010.02.033.
[23]	P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems,, Mathematical Surveys and Monographs, (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,, \emph{Ann. I. H. Poincar\'e - AN}, 29 (2012), 377. 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,, \emph{Appl. Math. Comput.}, 190 (2007), 1020. doi: 10.1016/j.amc.2006.11.187.
[26]	W. Niu, Long-time behavior for a nonlinear parabolic problem with variable exponents,, \emph{J. Math. Anal. Appl.}, 393 (2012), 56. doi: 10.1016/j.jmaa.2012.03.039.
[27]	K. Rajagopal and M. Růžička, Mathematical modelling of electrorheological materials,, \emph{Contin. Mech. Thermodyn.}, 13 (2001), 59. doi: 10.1007/s001610100034.
[28]	M. Růžička, Flow of shear dependent electrorheological fluids,, \emph{C. R. Acad. Sci. Paris Sér. I}, 329 (1999), 393. 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, (2000). doi: 10.1007/BFb0104029.
[30]	J. Simsen, A global attractor for a $p(x)$-Laplacian problem,, \emph{Nonlinear Anal.}, 73 (2010), 3278. doi: 10.1016/j.na.2010.06.087.
[31]	J. Simsen and C. B. Gentile, Well-posed $p$-laplacian problems with large diffusion,, \emph{Nonlinear Anal.}, 71 (2009), 4609. 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,, \emph{J. Math. Anal. Appl.}, 383 (2011), 71. 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,, \emph{J. Math. Anal. Appl.}, 398 (2013), 138. 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,, \emph{J. Math. Anal. Appl.}, 413 (2014), 685. 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,, \emph{J. Math. Anal. Appl.}, 342 (2008), 27. doi: 10.1016/j.jmaa.2007.11.046.
[36]	R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (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,, \emph{Nonlinear Anal.}, 72 (2010), 3887. 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,, \emph{Discrete Contin. Dyn. Syst. Series B}, 17 (2012), 2635. doi: 10.3934/dcdsb.2012.17.2635.
[39]	S. Yotsutani, Evolution equations associated with the subdifferentials,, \emph{J. Math. Soc. Japan}, 31 (1978), 623. doi: 10.2969/jmsj/03140623.

show all references

References:

[1]	R. Aboulaich, D. Meskine, A. Souissi, New diffusion models in image processing,, \emph{Computers and Mathematics with Applications}, 56 (2008), 874. 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,, \emph{Discrete Contin. Dyn. Syst. 2011, Suppl. Vol. I (2011), 22.
[3]	G. Akagi and K. Matsuura, Nonlinear diffusion equations driven by $p(\cdot)-$Laplacian,, \emph{NoDEA Nonlinear Differential Equations Appl.}, 20 (2013), 37. 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,, \emph{Nonlinear Anal.: Real World Applications}, 10 (2009), 2521. 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,, \emph{Nonlinear Analysis}, 60 (2005), 515. doi: 10.1016/j.na.2004.09.026.
[6]	S. N. Antontsev and S. Shmarev, Anisotropic parabolic equations with variable nonlinearity,, \emph{Publications Mathematiques}, 53 (2009), 355. 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,, \emph{J. Mathematical Sciences}, 150 (2008), 2289. 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 $\mathbbR^n$,, \emph{Math. Meth. Appl. Sci.}, 36 (2013), 1625. 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,, \emph{Nonlinear Anal.}, 72 (2010), 1967. 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, (2012). doi: 10.1007/978-1-4614-4581-4.
[13]	G. Chen and C. Zhong, Uniform attractors for non-autonomous $p-$Laplacian equations,, \emph{Nonlinear Anal.}, 68 (2008), 3349. doi: 10.1016/j.na.2007.03.025.
[14]	Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration,, \emph{SIAM J. Math.}, 66 (2006), 1383. doi: 10.1137/050624522.
[15]	L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents,, Springer-Verlag, (2011). doi: 10.1007/978-3-642-18363-8.
[16]	M. Effendiev, Attractors for Degenerate Parabolic type Equations,, Mathematical Surveys and Monographs, (2013).
[17]	F. Ettwein and M. Růžička, Existence of local strong solutions for motions of electrorheological fluids in three dimensions,, \emph{Computers and mathematics with applications}, 53 (2007), 595. 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,, \emph{Nonlinear Anal.}, 52 (2003), 1843. doi: 10.1016/S0362-546X(02)00150-5.
[19]	X. L. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega)$,, \emph{J. Math. Anal. Appl.}, 262 (2001), 749. doi: 10.1006/jmaa.2001.7618.
[20]	X. L. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$,, \emph{J. Math. Anal. Appl.}, 263 (2001), 424. 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,, \emph{Nonlinear Analysis: Real World Applications}, 12 (2011), 2904. 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,, \emph{Nonlinear Analysis}, 72 (2010), 4551. doi: 10.1016/j.na.2010.02.033.
[23]	P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems,, Mathematical Surveys and Monographs, (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,, \emph{Ann. I. H. Poincar\'e - AN}, 29 (2012), 377. 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,, \emph{Appl. Math. Comput.}, 190 (2007), 1020. doi: 10.1016/j.amc.2006.11.187.
[26]	W. Niu, Long-time behavior for a nonlinear parabolic problem with variable exponents,, \emph{J. Math. Anal. Appl.}, 393 (2012), 56. doi: 10.1016/j.jmaa.2012.03.039.
[27]	K. Rajagopal and M. Růžička, Mathematical modelling of electrorheological materials,, \emph{Contin. Mech. Thermodyn.}, 13 (2001), 59. doi: 10.1007/s001610100034.
[28]	M. Růžička, Flow of shear dependent electrorheological fluids,, \emph{C. R. Acad. Sci. Paris Sér. I}, 329 (1999), 393. 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, (2000). doi: 10.1007/BFb0104029.
[30]	J. Simsen, A global attractor for a $p(x)$-Laplacian problem,, \emph{Nonlinear Anal.}, 73 (2010), 3278. doi: 10.1016/j.na.2010.06.087.
[31]	J. Simsen and C. B. Gentile, Well-posed $p$-laplacian problems with large diffusion,, \emph{Nonlinear Anal.}, 71 (2009), 4609. 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,, \emph{J. Math. Anal. Appl.}, 383 (2011), 71. 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,, \emph{J. Math. Anal. Appl.}, 398 (2013), 138. 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,, \emph{J. Math. Anal. Appl.}, 413 (2014), 685. 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,, \emph{J. Math. Anal. Appl.}, 342 (2008), 27. doi: 10.1016/j.jmaa.2007.11.046.
[36]	R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (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,, \emph{Nonlinear Anal.}, 72 (2010), 3887. 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,, \emph{Discrete Contin. Dyn. Syst. Series B}, 17 (2012), 2635. doi: 10.3934/dcdsb.2012.17.2635.
[39]	S. Yotsutani, Evolution equations associated with the subdifferentials,, \emph{J. Math. Soc. Japan}, 31 (1978), 623. doi: 10.2969/jmsj/03140623.

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