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February  2019, 24(2): 529-546. doi: 10.3934/dcdsb.2018194

The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition

Wen Tan 1,2,

1. 

School of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, Guangdong, China
2. 

Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province, College of Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, Guangdong, China

Received  June 2016 Revised  March 2018 Published  February 2019 Early access  June 2018

Fund Project: The author was supported by NSF of China (Grant No. 11471148, Grant No. 11522109, Grant No. 11571240) and China Postdoctoral Science Foundation (Grant No. 2018M633101).

In this paper, we investigate the asymptotic regularity of the minimal pullback attractor of a non-autonomous quasi-linear parabolic $p$-Laplacian equation with dynamical boundary condition. First, we establish the higher-order integrability of the difference of solutions near the initial time. Then we show that, under the assumption that the time-depending forcing terms only satisfy some $L^2$ integrability, the $L^2(Ω)× L^2(\partialΩ)$ pullback $\mathscr{D}$-attractor can actually attract the $L^2(Ω)× L^2(\partialΩ)$-bounded set in $L^{2+δ}(Ω)× L^{2+δ}(\partialΩ)$-norm for any $δ∈[0,∞)$.

Keywords: Pullback attractor, asymptotic regularity, dynamical boundary condition, higher-order attraction.
Mathematics Subject Classification: Primary: 35K57, 35B40, 35B41.
Citation: Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194
References:
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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

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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

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I. Chueshov and B. Schmalfuß, Parabolic stochastic partial differential equations with dynamical boundary conditions, Differential Integral Equations, 17 (2004), 751-780.   Google Scholar

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H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

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C. G. Gal, On a class of degenerate parabolic equations with dynamic boundary conditions, J. Differential Equations, 253 (2012), 126-166.  doi: 10.1016/j.jde.2012.02.010.  Google Scholar

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C. G. Gal and M. Meyries, Nonlinear elliptic problems with dynamical boundary conditions of reactive and reactive-diffusive type, Proc. Lond. Math. Soc., 108 (2014), 1351-1380.  doi: 10.1112/plms/pdt057.  Google Scholar

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C. G. Gal and M. Warma, Well-posedness and long term behavior of quasilinear parabolic equations with nonlinear dynamic boundary conditions, Differential Integral Equations, 23 (2010), 327-358.   Google Scholar

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P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, Vol. 176, 2011. doi: 10.1090/surv/176.  Google Scholar

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G. Łukaszewicz, On pullback attractors in $H_0^1$ for nonautonomous reaction-diffusion equations, Internat. J. Bifur. Chaos, 20 (2010), 2637-2644.  doi: 10.1142/S0218127410027258.  Google Scholar

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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

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A. Rodrguez-Bernal, Attractors for parabolic equations with nonlinear boundary conditions, critical exponents and singular initial data, J. Differential Equations, 181 (2002), 165-196.  doi: 10.1006/jdeq.2001.4072.  Google Scholar

[21]

C. Sun and Y. Yuan, $L^p$-type pullback attractors for a semilinear heat equation on time-varying domains, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1029-1052.  doi: 10.1017/S0308210515000177.  Google Scholar

[22]

L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lecture Notes of the Unione Matematica Italiana, Springer, 2007.  Google Scholar

[23]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[24]

L. Yang and M. Yang, Long-time behavior of reaction-diffusion equations with dynamical boundary condition, Nonlinear Anal., 74 (2011), 3876-3883.  doi: 10.1016/j.na.2011.02.022.  Google Scholar

[25]

L. Yang, M. Yang and P. E. Kloeden, Pullback attractors for non-autonomous quasilinear parabolic equations with dynamical boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2635-2651.  doi: 10.3934/dcdsb.2012.17.2635.  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), 608-618.  doi: 10.1016/j.jmaa.2011.05.046.  Google Scholar

[2]

J. M. Arriteta, P. Quittner and A. Rodrguez-Bernal, Parabolic problems with nonlinear dynamical boundary conditions and singular initial data, Diffential Integral Equations, 14 (2001), 1487-1510.   Google Scholar

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, 2011.  Google Scholar

[4]

D. Cao, C. Sun and M. Yang, Dynamical for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872.  doi: 10.1016/j.jde.2015.02.020.  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]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[7]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ., Vol. 49, Amer. Math. Soc., Providence, RI, 2002.  Google Scholar

[8]

J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, 2000. doi: 10.1017/CBO9780511526404.  Google Scholar

[9]

I. Chueshov and B. Schmalfuß, Parabolic stochastic partial differential equations with dynamical boundary conditions, Differential Integral Equations, 17 (2004), 751-780.   Google Scholar

[10]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[11]

C. G. Gal, On a class of degenerate parabolic equations with dynamic boundary conditions, J. Differential Equations, 253 (2012), 126-166.  doi: 10.1016/j.jde.2012.02.010.  Google Scholar

[12]

C. G. Gal and M. Meyries, Nonlinear elliptic problems with dynamical boundary conditions of reactive and reactive-diffusive type, Proc. Lond. Math. Soc., 108 (2014), 1351-1380.  doi: 10.1112/plms/pdt057.  Google Scholar

[13]

C. G. Gal and M. Warma, Well-posedness and long term behavior of quasilinear parabolic equations with nonlinear dynamic boundary conditions, Differential Integral Equations, 23 (2010), 327-358.   Google Scholar

[14]

C. G. Gal and J. Shomberg, Coleman-Gurtin type equations with dynamic boundary conditions, Phys. D, 292 (2015), 29-45.  doi: 10.1016/j.physd.2014.10.008.  Google Scholar

[15]

G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480.   Google Scholar

[16]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, Vol. 176, 2011. doi: 10.1090/surv/176.  Google Scholar

[17]

G. Leoni, A First Course in Sobolev Spaces, Grad. Stud. Math., vol. 105, Amer. Math. Soc., 2009. doi: 10.1090/gsm/105.  Google Scholar

[18]

G. Łukaszewicz, On pullback attractors in $H_0^1$ for nonautonomous reaction-diffusion equations, Internat. J. Bifur. Chaos, 20 (2010), 2637-2644.  doi: 10.1142/S0218127410027258.  Google Scholar

[19]

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

[20]

A. Rodrguez-Bernal, Attractors for parabolic equations with nonlinear boundary conditions, critical exponents and singular initial data, J. Differential Equations, 181 (2002), 165-196.  doi: 10.1006/jdeq.2001.4072.  Google Scholar

[21]

C. Sun and Y. Yuan, $L^p$-type pullback attractors for a semilinear heat equation on time-varying domains, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1029-1052.  doi: 10.1017/S0308210515000177.  Google Scholar

[22]

L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lecture Notes of the Unione Matematica Italiana, Springer, 2007.  Google Scholar

[23]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[24]

L. Yang and M. Yang, Long-time behavior of reaction-diffusion equations with dynamical boundary condition, Nonlinear Anal., 74 (2011), 3876-3883.  doi: 10.1016/j.na.2011.02.022.  Google Scholar

[25]

L. Yang, M. Yang and P. E. Kloeden, Pullback attractors for non-autonomous quasilinear parabolic equations with dynamical boundary conditions, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2635-2651.  doi: 10.3934/dcdsb.2012.17.2635.  Google Scholar

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