非经典扩散方程的拉回吸引子在H01(Ω)中的上半连续性

Abstract

Abstract:

This paper is concerned with upper semicontinuity of pullback attractors, for a nonautonomous nonclassical diffusion equation with critical nonlinearity. In particular, under some proper assumptions, we prove that, the pullback attractor {A_ε(t)}_t∈R of equation (1.1) with ε∈[0,1] satisfies that for any [a,b]⊂R and ε₀∈[0,1], dist_H₀¹(A_ε(t),A_ε₀(t))=0 and ∪_t∈[a,b]∪_ε∈[0,1]A_ε(t) is precompact in H₀¹(Ω).

Key words: Pullback attractor, Upper semicontinuity, Nonclassical diffusion equation

CLC Number:

O175

Wang Yonghai, Qin Yuming. Upper Semicontinuity of Pullback Attractors for Nonclassical Diffusion Equations in H₀¹(Ω)[J].Acta mathematica scientia,Series A, 2016, 36(5): 946-957.

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References

[1] Aifantis E C. On the problem of diffusion in solids. Acta Mech, 1980, 37: 265-296
[2] Marín-Rubio M, Real J. Regularity results and exponential growth for pullback attractors of a non-autonomous reaction-diffusion model with dynamical boundary conditions. Nonlinear Analysis: Real World Applications, 2014, 20: 112-125
[3] Anh C T, Bao T Q. Pullback attractors for a class of non-autonomous nonclassical diffusion equation. Nonlinear Analysis, 2010, 40: 399-412
[4] Arrieta J M, Carvalho A N, Langa J A, Rodriguez-Bernal. Continuity of dynamical structures for nonautonomous evolution equations under singular perturbations. J Dyn Diff Equa, 2012, 24: 427-481
[5] Babin A V, Vishik M I. Attractors of Evolution Equations. Amsterdam: North-Holland, 1992
[6] Caraballo T, Langa J A. On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems. Dyn Contin Discrete Impuls Syst Ser A Math Anal, 2003, 10: 491-513
[7] Caraballo T, Langa J A, Robinson J C. Upper semicontinuity of attractors for small random perturbations of dynamical systems. Commun Partial Diff Eqns, 1998, 23: 1557-1581
[8] Caraballo T, Herreta-Cobos M, Marín-Rubio P. Long-time behavior of a non-autonomous parabolic equation with nonlocal diffusion and sublinear terms. Nonlinear Analysis, 2015, 121: 3-18
[9] Caraballo T, Lukaszewicz G, Real J. Pullback attractors for asymptotically compact non-autonomous dynamical systems. Nonlinear Anal, 2006, 64: 484-498
[10] Carvalho A N, Langa J A, Robinson J C. On the continuity of pullback attractors for evolution processes. Nonlinear Anal, 2009, 71: 1812-1824
[11] Carvalho A N, Langa J A. Non-autonomous perturbation of autonomous semilinear differential equations: continuity of local stable and unstable manifolds. J Differential Equations, 2007, 223: 622-653
[12] Chepyzhov V V, Vishik M I. Attractors for Equations of Mathematical Physics. Providence, RI: Amer Math Soc Colloq Pub, 2002
[13] Elliot C M, Kostin I N. Lower semicontinuity of a nonhyperbolic attractor for the viscous Cahn-Hilliard equation. Nonlinearity, 1996, 9: 678-702
[14] Garcia-Luengo J, Marín-Rubio P, Real J. Pullback attractors in V for non-autonomous 2D-Navier-Stokes equations and their tempered behaviour. J Differential Equations, 2012, 252: 4333-4356
[15] Hale J K, Raugel G. Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation. J Differential Equations, 1988, 73: 197-214
[16] Hale J K, Raugel G. Lower semicontinuity of the attractor for a singularly perturbed hyperbolic equation. J Dyn Diff Equa, 1990, 2: 19-67
[17] Kapitanski L V, Kostin I N. Attractors of nonlinear evolution equations and their approximations. Leningrad Math J, 1991, 2: 97-117
[18] Kloeden P E, Langa J A. Flattening, squeezing and the existence of random attractors. Proc Roy Soc London A, 2007, 463: 163-181
[19] Kostin I N. Lower semicontinuity of a non-hyperbolic attractor. J London Math Soc, 1995, 52: 568-582
[20] Peter J C, Gurtin M E. On a theory of heat conduction involving two temperatures. Z Angew Math Phys, 1968, 19: 614-627
[21] Qin Y M, Li H Y. Global existence and exponential stability of solutions to the quasilinear thermo-diffusion equations with second sound. Acta Mathematica Scientia, 2014, 34B(3): 758-778
[22] Robinson J C. Stability of random attractors under perturbation and approximation. J Differential Equations, 2002, 186: 652-669
[23] Robinson J C. Infinite-Dimensional Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors. Cambridge: Cambridge University, 2001
[24] Sun C Y, Yang M H. Dynamics of the nonclassical diffusion equations. Asymptotic Anal, 2008, 59: 51-81
[25] Truesdell C, Noll W. The Nonlinear Field Theories of Mechanics. Berlin: Springer, 1995
[26] Wang B X. Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems. J Differential Equations, 2012, 253: 1544-1583
[27] Wang S Y, Li D S, Zhong C K. On the dynamics of a class of nonclassical parabolic equations. J Math Anal Appl, 2006, 317: 565-582
[28] Zelik S. Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent. Commun Pure Appl Anal, 2004, 3: 921-934

Upper Semicontinuity of Pullback Attractors for Nonclassical Diffusion Equations in H₀¹(Ω)

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Upper Semicontinuity of Pullback Attractors for Nonclassical Diffusion Equations in H01(Ω)

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Upper Semicontinuity of Pullback Attractors for Nonclassical Diffusion Equations in H₀¹(Ω)