# American Institute of Mathematical Sciences

October  2012, 17(7): 2635-2651. doi: 10.3934/dcdsb.2012.17.2635

## Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions

 1 School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China 2 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China 3 Institut für Mathematik, Johann Wolfgang Goethe Universität, D-60054 Frankfurt am Main, Germany

Received  April 2012 Revised  May 2012 Published  July 2012

The existence of a unique minimal pullback attractor is established for the evolutionary process associated with a non-autonomous quasi-linear parabolic equations with a dynamical boundary condition in $L^{r_1}(\Omega)\times L^{r_1}(\Gamma)$ under that assumption that the external forcing term satisfies a weak integrability condition, where $r_1$ $>$ $2$ is determined by the order of the nonlinearity.
Citation: Lu Yang, Meihua Yang, Peter E. Kloeden. Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2635-2651. doi: 10.3934/dcdsb.2012.17.2635
##### References:
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A. Langa and B. Schmalfuß, Finite dimensionality of attractors for nonautonomous dynamical systems given by partial differential equations, Stoch. Dyn., 4 (2004), 385-404.  Google Scholar [26] Y. J. Li and C. K. 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.  Google Scholar [27] Y. J. Li, S. Y. Wang and H. Q. Wu, Pullback attractors for non-autonomous reaction-diffusion equations in $L^p$, Appl. Math. Comput., 207 (2009), 373-379. doi: 10.1016/j.amc.2008.10.065.  Google Scholar [28] G. Łukaszewicz and A. Tarasińska, On $H^1$-pullback attractors for nonautonomous micropolar fluid equations in a bounded domain, Nonlinear Anal., 71 (2009), 782-788. doi: 10.1016/j.na.2008.10.124.  Google Scholar [29] G. Łukaszewicz, On pullback attractors in $H^1_0$ for nonautonomous reaction-diffusion equations, Internat. J. Bifur. Chaos, 20 (2010), 2637-2644. doi: 10.1142/S0218127410027258.  Google Scholar [30] G. Łukaszewicz, On pullback attractors in $L^p$ for nonautonomous reaction-diffusion equations, Nonlinear Anal., 73 (2010), 350-357. doi: 10.1016/j.na.2010.03.023.  Google Scholar [31] J. C. Robinson, "Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors," Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001.  Google Scholar [32] H. T. Song, Pullback attractors of non-autonomous reaction-diffusion equations in $H^1_0$, J. Differential Equations, 249 (2010), 2357-2376. doi: 10.1016/j.jde.2010.07.034.  Google Scholar [33] H. T. Song and H. Q. Wu, Pullback attractors of non-autonomous reaction-diffusion equations, J. Math. Anal. Appl., 325 (2007), 1200-1215. doi: 10.1016/j.jmaa.2006.02.041.  Google Scholar [34] J. Sprekels and H. Wu, A note on parabolic equation with nonlinear dynamic boundary condition, Nonlinear Anal., 72 (2010), 3028-3048. doi: 10.1016/j.na.2009.11.043.  Google Scholar [35] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.  Google Scholar [36] H. Wu, Convergence to equilibrium for the semilinear parabolic equation with dynamic boundary condition, Adv. Math. Sci. Appl., 17 (2007), 67-88.  Google Scholar [37] L. Yang, Uniform attractors for the closed process and applications to the reaction-diffusion equation with dynamical boundary condition, Nonlinear Anal., 71 (2009), 4012-4025. doi: 10.1016/j.na.2009.02.083.  Google Scholar [38] C-K. Zhong, M. H. Yang and C. Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399. doi: 10.1016/j.jde.2005.06.008.  Google Scholar

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##### References:
 [1] M. Anguiano, P. Marín-Rubio and J. Real, Pullback attractors for non-autonomous reaction-diffusion equations with dynamical boundary conditions, J. Math. Anal. Appl., 383 (2011), 608-618. doi: 10.1016/j.jmaa.2011.05.046.  Google Scholar [2] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar [3] T. Caraballo, P. E. Kloeden and J. Real, Pullback and forward attractors for a damped wave equation with delays, Stoch. Dyn., 4 (2004), 405-423.  Google Scholar [4] T. Caraballo, J. A. Langa and J. Valero, The dimension of attractors of nonautonomous partial differential equations, ANZIAM. J., 45 (2003), 207-222. doi: 10.1017/S1446181100013274.  Google Scholar [5] T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111.  Google Scholar [6] T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, C.R. Acad. Sci. Paris, Ser., 342 (2006), 263-268.  Google Scholar [7] C. Cavaterra, C. G. Gal, M. Grasselli and A. Miranville, Phase-field systems with nonlinear coupling and dynamical boundary conditions, Nonlinear Anal., 72 (2010), 2375-2399. doi: 10.1016/j.na.2009.11.002.  Google Scholar [8] D. N. Cheban, P. E. Kloeden and B. Schmalfuß, The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems, Nonlinear Dyn. Syst. Theory, 2 (2002), 125-144.  Google Scholar [9] V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," American Mathematical Society Colloquium Publications, 49, Amer. Math. Soc., Providence, RI, 2002.  Google Scholar [10] I. Chueshov and B. Schmalfuß, Parabolic stochastic partial differential equations with dynamical boundary conditions, Differential Integral Equations, 17 (2004), 751-780.  Google Scholar [11] I. Chueshov and B. Schmalfuß, Qualitative behavior of a class of stochastic parabolic PDEs with dynamical boundary conditions, Discrete Contin. Dyn. Syst., 18 (2007), 315-338. doi: 10.3934/dcds.2007.18.315.  Google Scholar [12] J. W. Cholewa and T. Dlotko, "Global Attractors in Abstract Parabolic Problems," London Mathematical Society Lecture Note Series, 278, Cambridge University Press, Cambridge, 2000.  Google Scholar [13] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.  Google Scholar [14] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Differential Equations, 9 (1997), 307-341.  Google Scholar [15] J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations, 18 (1993), 1309-1364. doi: 10.1080/03605309308820976.  Google Scholar [16] Z.H. Fan and C.K. Zhong, Attractors for parabolic equations with dynamic boundary conditions, Nonlinear Anal., 68 (2008), 1723-1732. doi: 10.1016/j.na.2007.01.005.  Google Scholar [17] C. G. Gal and M. Grasselli, The non-isothermal Allen-Cahn equation with dynamic boundary conditions, Discrete Contin. Dyn. Syst., 22 (2008), 1009-1040. doi: 10.3934/dcds.2008.22.1009.  Google Scholar [18] C. G. Gal and M. Warma, Well-posedness and the global attractor of some quasi-linear parabolic equations with nonlinear dynamic boundary conditions, Differential Integral Equations, 23 (2010), 327-358.  Google Scholar [19] C. G. Gal, On a class of degenerate parabolic equations with dynamical boundary conditions,, \arXiv{1109.0469}., ().   Google Scholar [20] M. Grasselli, A. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamical boundary conditions and singular potentials, Discrete Contin. Dyn. Syst., 28 (2010), 67-98. doi: 10.3934/dcds.2010.28.67.  Google Scholar [21] P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112.  Google Scholar [22] P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753.  Google Scholar [23] P. E. Kloeden and B. Schmalfuß, Asymptotic behaviour of nonautonomous difference inclusions, Systems Control Lett., 33 (1998), 275-278. doi: 10.1016/S0167-6911(97)00107-2.  Google Scholar [24] J. A. Langa, G. Łukaszewicz and J. Real, Finite fractal dimension of pullback attractors for non-autonomous 2D Navier-Stokes equations in some unbounded domains, Nonlinear Anal., 66 (2007), 735-749. doi: 10.1016/j.na.2005.12.017.  Google Scholar [25] J. A. Langa and B. Schmalfuß, Finite dimensionality of attractors for nonautonomous dynamical systems given by partial differential equations, Stoch. Dyn., 4 (2004), 385-404.  Google Scholar [26] Y. J. Li and C. K. 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.  Google Scholar [27] Y. J. Li, S. Y. Wang and H. Q. Wu, Pullback attractors for non-autonomous reaction-diffusion equations in $L^p$, Appl. Math. Comput., 207 (2009), 373-379. doi: 10.1016/j.amc.2008.10.065.  Google Scholar [28] G. Łukaszewicz and A. Tarasińska, On $H^1$-pullback attractors for nonautonomous micropolar fluid equations in a bounded domain, Nonlinear Anal., 71 (2009), 782-788. doi: 10.1016/j.na.2008.10.124.  Google Scholar [29] G. Łukaszewicz, On pullback attractors in $H^1_0$ for nonautonomous reaction-diffusion equations, Internat. J. Bifur. Chaos, 20 (2010), 2637-2644. doi: 10.1142/S0218127410027258.  Google Scholar [30] G. Łukaszewicz, On pullback attractors in $L^p$ for nonautonomous reaction-diffusion equations, Nonlinear Anal., 73 (2010), 350-357. doi: 10.1016/j.na.2010.03.023.  Google Scholar [31] J. C. Robinson, "Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors," Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001.  Google Scholar [32] H. T. Song, Pullback attractors of non-autonomous reaction-diffusion equations in $H^1_0$, J. Differential Equations, 249 (2010), 2357-2376. doi: 10.1016/j.jde.2010.07.034.  Google Scholar [33] H. T. Song and H. Q. Wu, Pullback attractors of non-autonomous reaction-diffusion equations, J. Math. Anal. Appl., 325 (2007), 1200-1215. doi: 10.1016/j.jmaa.2006.02.041.  Google Scholar [34] J. Sprekels and H. Wu, A note on parabolic equation with nonlinear dynamic boundary condition, Nonlinear Anal., 72 (2010), 3028-3048. doi: 10.1016/j.na.2009.11.043.  Google Scholar [35] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.  Google Scholar [36] H. Wu, Convergence to equilibrium for the semilinear parabolic equation with dynamic boundary condition, Adv. Math. Sci. Appl., 17 (2007), 67-88.  Google Scholar [37] L. Yang, Uniform attractors for the closed process and applications to the reaction-diffusion equation with dynamical boundary condition, Nonlinear Anal., 71 (2009), 4012-4025. doi: 10.1016/j.na.2009.02.083.  Google Scholar [38] C-K. Zhong, M. H. Yang and C. Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399. doi: 10.1016/j.jde.2005.06.008.  Google Scholar
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