# 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:
 [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. 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 (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. 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. 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. 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, 342 (2006), 263. 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. 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. Google Scholar [9] V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics,", American Mathematical Society Colloquium Publications, 49 (2002). Google Scholar [10] I. Chueshov and B. Schmalfuß, Parabolic stochastic partial differential equations with dynamical boundary conditions,, Differential Integral Equations, 17 (2004), 751. 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. 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 (2000). Google Scholar [13] H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Theory Related Fields, 100 (1994), 365. doi: 10.1007/BF01193705. Google Scholar [14] H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Differential Equations, 9 (1997), 307. Google Scholar [15] J. Escher, Quasilinear parabolic systems with dynamical boundary conditions,, Comm. Partial Differential Equations, 18 (1993), 1309. 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. 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. 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. 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. doi: 10.3934/dcds.2010.28.67. Google Scholar [21] P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems,, Stoch. Dyn., 3 (2003), 101. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (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. 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. 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. doi: 10.1016/j.na.2009.11.043. Google Scholar [35] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Second edition, 68 (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. 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. 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. doi: 10.1016/j.jde.2005.06.008. Google Scholar

show all references

##### 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. 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 (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. 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. 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. 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, 342 (2006), 263. 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. 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. Google Scholar [9] V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics,", American Mathematical Society Colloquium Publications, 49 (2002). Google Scholar [10] I. Chueshov and B. Schmalfuß, Parabolic stochastic partial differential equations with dynamical boundary conditions,, Differential Integral Equations, 17 (2004), 751. 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. 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 (2000). Google Scholar [13] H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Theory Related Fields, 100 (1994), 365. doi: 10.1007/BF01193705. Google Scholar [14] H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Differential Equations, 9 (1997), 307. Google Scholar [15] J. Escher, Quasilinear parabolic systems with dynamical boundary conditions,, Comm. Partial Differential Equations, 18 (1993), 1309. 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. 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. 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. 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. doi: 10.3934/dcds.2010.28.67. Google Scholar [21] P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems,, Stoch. Dyn., 3 (2003), 101. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (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. 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. 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. doi: 10.1016/j.na.2009.11.043. Google Scholar [35] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Second edition, 68 (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. 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. 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. doi: 10.1016/j.jde.2005.06.008. Google Scholar
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