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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 |
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 $δ∈[0,∞)$.
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] |
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.
|
[3] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, 2011. |
[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. |
[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] |
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. 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. |
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] |
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.
|
[3] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, 2011. |
[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. |
[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] |
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. 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. |
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