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

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  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 $δ∈[0,∞)$.

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:
[1]

M. AnguianoP. 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]

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

[3]

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

[4]

D. CaoC. 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.

[5]

T. CaraballoG. 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]

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.

[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.

[8]

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

[9]

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

[10]

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

[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.

[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.

[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.

[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.

[15]

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

[16]

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

[17]

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

[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.

[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.

[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.

[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.

[22]

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

[23]

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

[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.

[25]

L. YangM. 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.

show all references

References:
[1]

M. AnguianoP. 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]

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

[3]

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

[4]

D. CaoC. 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.

[5]

T. CaraballoG. 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]

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.

[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.

[8]

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

[9]

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

[10]

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

[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.

[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.

[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.

[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.

[15]

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

[16]

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

[17]

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

[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.

[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.

[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.

[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.

[22]

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

[23]

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

[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.

[25]

L. YangM. 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.

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