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

Lu Yang; Meihua Yang; Peter E. Kloeden

doi:10.3934/dcdsb.2012.17.2635

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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

Lu Yang ^1, , Meihua Yang ^2, and Peter E. Kloeden ^3,

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

Abstract
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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.

Keywords: quasi-linear parabolic equation, p-Laplacian, dy- namical boundary condition., Pullback attractor.

Mathematics Subject Classification: 37L05, 35B40, 35B4.

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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[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
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[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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