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

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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 and 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), 608-618. doi: 10.1016/j.jmaa.2011.05.046.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	I. Chueshov and B. Schmalfuß, Parabolic stochastic partial differential equations with dynamical boundary conditions, Differential Integral Equations, 17 (2004), 751-780.
[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.
[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.
[13]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.
[14]	H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Differential Equations, 9 (1997), 307-341.
[15]	J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations, 18 (1993), 1309-1364. doi: 10.1080/03605309308820976.
[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.
[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.
[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.
[19]	C. G. Gal, On a class of degenerate parabolic equations with dynamical boundary conditions, arXiv:1109.0469.
[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.
[21]	P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[35]	R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.
[36]	H. Wu, Convergence to equilibrium for the semilinear parabolic equation with dynamic boundary condition, Adv. Math. Sci. Appl., 17 (2007), 67-88.
[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.
[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.

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), 608-618. doi: 10.1016/j.jmaa.2011.05.046.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	I. Chueshov and B. Schmalfuß, Parabolic stochastic partial differential equations with dynamical boundary conditions, Differential Integral Equations, 17 (2004), 751-780.
[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.
[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.
[13]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.
[14]	H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Differential Equations, 9 (1997), 307-341.
[15]	J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations, 18 (1993), 1309-1364. doi: 10.1080/03605309308820976.
[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.
[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.
[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.
[19]	C. G. Gal, On a class of degenerate parabolic equations with dynamical boundary conditions, arXiv:1109.0469.
[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.
[21]	P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[35]	R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.
[36]	H. Wu, Convergence to equilibrium for the semilinear parabolic equation with dynamic boundary condition, Adv. Math. Sci. Appl., 17 (2007), 67-88.
[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.
[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.

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