Time-dependent global attractor for a class of nonclassical parabolic equations. (English) Zbl 1442.35223

Summary: Based on the recent theory of time-dependent global attractors in the works of M. Conti et al. [J. Differ. Equations 255, No. 6, 1254–1277 (2013; Zbl 1288.35098)] and F. Di Plinio et al. [Discrete Contin. Dyn. Syst. 29, No. 1, 141–167 (2011; Zbl 1223.37100)], we prove the existence of time-dependent global attractors as well as the regularity of the time-dependent global attractor for a class of nonclassical parabolic equations.


35K58 Semilinear parabolic equations
35B41 Attractors
35K35 Initial-boundary value problems for higher-order parabolic equations
Full Text: DOI


[1] Aifantis, E. C., On the problem of diffusion in solids, Acta Mechanica, 37, 3-4, 265-296 (1980) · Zbl 0447.73002
[2] Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physic (1997), New York, NY, USA: Springer, New York, NY, USA
[3] Lions, J. L.; Magenes, E., Non-Homogeneous Boundary Value Problems and Applications (1972), Berlin, Germany: Spring, Berlin, Germany
[4] Kuttler, K.; Aifantis, E. C., Existence and uniqueness in nonclassical diffusion, Quarterly of Applied Mathematics, 45, 3, 549-560 (1987)
[5] Kuttler, K.; Aifantis, E., Quasilinear evolution equations in nonclassical diffusion, SIAM Journal on Mathematical Analysis, 19, 1, 110-120 (1988) · Zbl 0675.35053
[6] Aifantis, E. C., Gradient nanomechanics: applications to deformation, fracture, and diffusion in nanopolycrystals, Metallurgical and Materials Transactions A, 42, 10, 2985-2998 (2011)
[7] Kalantarov, V. K., On the attractors for some non-linear problems of mathematical physics, Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta imeni V. A. Steklova Akademii Nauk SSSR (LOMI), 152, 50-54 (1986)
[8] Xiao, Y., Attractors for a nonclassical diffusion equation, Acta Mathematicae Applicatae Sinica, 18, 2, 273-276 (2002) · Zbl 1017.35025
[9] Sun, C. Y.; Wang, S. Y.; Zhong, C. K., Global attractors for a nonclassical diffusion equation, Acta Mathematica Sinica, English Series, 23, 7, 1271-1280 (2007)
[10] Wang, S. Y.; Li, D. S.; Zhong, C. K., On the dynamics of a class of nonclassical parabolic equations, Journal of Mathematical Analysis and Applications, 317, 2, 565-582 (2006) · Zbl 1092.35016
[11] Sun, C.; Yang, M., Dynamics of the nonclassical diffusion equations, Asymptotic Analysis, 59, 1-2, 51-81 (2008)
[12] Liu, Y. F.; Ma, Q. Z., Exponential attractors for a nonclassical diffusion equation, Electronic Journal of Differential Equations, 9, 1-9 (2009)
[13] Ma, Q. Z.; Liu, Y. F.; Zhang, F. H., Global attractors in H1(RN) for nonclassical diffusion equations, Discrete Dynamics in Nature and Society, 2012 (2012) · Zbl 1258.35037
[14] Wu, H. Q.; Zhang, Z. Y., Asymptotic regularity for the nonclassical diffusion equation with lower regular forcing term, Dynamical Systems, 26, 4, 391-400 (2011) · Zbl 1229.35218
[15] Pan, L.-X.; Liu, Y.-F., Robust exponential attractors for the non-autonomous nonclassical diffusion equation with memory, Dynamical Systems, 28, 4, 501-517 (2013) · Zbl 1286.35048
[16] Zhang, F.-H.; Liu, Y.-F., Pullback attractors in H1(RN) for non-autonomous nonclassical diffusion equations, Dynamical Systems, 29, 1, 106-118 (2014) · Zbl 1284.35084
[17] Conti, M.; Pata, V.; Temam, R., Attractors for processes on time-dependent spaces. Applications to wave equations, Journal of Differential Equations, 255, 6, 1254-1277 (2013) · Zbl 1288.35098
[18] Caraballo, T.; Garrido-Atienza, M. J.; Schmalfuss, B., Existence of exponentially attracting stationary solutions for delay evolution equations, Discrete and Continuous Dynamical Systems B, 18, 2-3, 271-293 (2007) · Zbl 1125.60058
[19] Caraballo, T.; Kloeden, P. E.; Schmalfu√ü, B., Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Applied Mathematics and Optimization, 50, 3, 183-207 (2004) · Zbl 1066.60058
[20] Flandoli, F.; Schmalfuss, B., Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stochastics and Stochastics Reports, 59, 1-2, 21-45 (1996)
[21] di Plinio, F.; Duane, G. S.; Temam, R., Time-dependent attractor for the oscillon equation, Discrete and Continuous Dynamical Systems A, 29, 1, 141-167 (2011) · Zbl 1223.37100
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