Pullback attractors for non-autonomous reaction-diffusion equations on {ℝ}^n

Abstract

We study the long time behavior of solutions of the non-autonomous reaction-diffusion equation defined on the entire space Rn when external terms are unbounded in a phase space. The existence of a pullback global attractor for the equation is established in L2(Rn) and H1(Rn), respectively. The pullback asymptotic compactness of solutions is proved by using uniform a priori estimates on the tails of solutions outside bounded domains.

Key words： Pullback attractor; asymptotic compactness; non-autonomous equation

References

1 Antoci F, Prizzi M. Attractors and global averaging of non-autonomous reactiondiffusion equations in ℝn. Topological Methods in Nonlinear Analysis, 2002, 20: 229-259

2 Arnold L. Random Dynamical Systems. Berlin: Springer-Verlag, 1998

3 Aulbach B, Rasmussen M, Siegmund S. Approximation of attractors of nonautonomous dynamical systems. Discrete Contin Dyn Syst B, 2005, 5: 215-238

4 Ball J M. Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations. J Nonl Sci, 1997, 7: 475-502

5 Ball J M. Global attractors for damped semilinear wave equations. Discrete Contin Dyn Syst, 2004, 10: 31-52

6 Bates P W, Lisei H, Lu K. Attractors for stochastic lattice dynamical systems. Stoch Dyn, 2006, 6: 1-21

7 Bates P W, Lu K, Wang B. Random attractors for stochastic reaction-diffusion equations on unbounded domains. J Differential Equations, 2009, 246: 845-869

8 Caraballo T, Kloeden P E, Real J. Pullback and forward attractors for a damped wave equation with delays. Stochastic Dyn, 2004, 4: 405-423

9 Caraballo T, Langa J A, Robinson J C. A stochastic pitchfork bifurcation in a reaction-diffusion equation. Proc R Soc Lond, A, 2001, 457: 2041-2061

10 Caraballo T, Lukaszewicz G, Real J. Pullback attractors for asymptotically compact non-autonomous dynamical systems. Nonlinear Anal, 2006, 64: 484-498

11 Caraballo T, Lukaszewicz G, Real J. Pullback attractors for non-autonomous 2DNavier- Stokes equations in some unbounded domains. C R Acad Sci Paris I, 2006, 342: 263-268

12 Caraballo T, Real J. Attractors for 2D-Navier-Stokes models with delays. J Differential Equations, 2004, 205: 271-297

13 Cheban D N. Global Attractors of Non-Autonomous Dissipative Dynamical Systems. Singapore: World Scientific, 2004

14 Cheban D N, Duan J. Almost periodic motions and global attractors of the nonautonomous Navier-Stokes equations. J Dyn Diff Eqns, 2004, 16: 1-34

15 Cheban D N, Kloeden P E, Schmalfuss B. The relationship between pullback, forward and global attractors of nonautonomous dynamical systems. Nonlin Dyn Syst Theory, 2002, 2: 9-28

16 Chepyzhov V V, Vishik M I. Attractors of non-autonomous dynamical systems and their dimensions. J Math Pures Appl, 1994, 73: 279-333

17 Chueshov I D. Monotone Random Systems—Theory and Applications. Lecture Notes in Mathematics, Vol 1779. Berlin: Springer-Verlag, 2002

18 Flandoli F, Schmalfuβ B. Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise. Stoch Stoch Rep, 1996, 59: 21-45

19 Goubet O, Rosa R. Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line. J Differential Equations, 2002, 185: 25-53

20 Haraux A. Attractors of asymptotically compact processes and applications to nonlinear partial differential equations. Commun Partial Differential Equations, 1988, 13: 1383-1414

21 Ju N. The H¹-compact global attractor for the solutions to the Navier-Stokes equations in two-dimensional unbounded domains. Nonlinearity, 2000, 13: 1227-1238

22 Langa J A, Lukaszewicz G, Real J. Finite fractal dimension of pullback attractors for non-autonomous 2D Navier-Stokes equations in some unbounded domains. Nonlin Anal, 2007, 66: 735-749

23 Langa J A, Schmalfuss B. Finite dimensionality of attractors for non-autonomous dynamical systems given by partial differential equations. Stochastic Dyn, 2004, 3: 385-404

24 Lu K, Wang B. Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains. J Differential Equations, 2001, 170: 281-316

25 Lu S, Wu H, Zhong C. Attractors for non-autonomous 2D Navier-Stokes equations with normal external forces. Discrete Continuous Dynamical Systems A, 2005, 13: 701-719

26 Moise I, Rosa R. On the regularity of the global attractor of a weakly damped, forced Korteweg-de Vries equation. Adv Differential Equations, 1997, 2: 257-296

27 Moise I, Rosa R, Wang X. Attractors for noncompact nonautonomous systems via energy equations. Discrete Continuous Dynamical Systems, 2004, 10: 473-496

28 Prizzi M. Averaging, Conley index continuation and recurrent dynamics in almostperiodic parabolic equations. J Differential Equations, 2005, 210: 429-451

29 Rosa R. The global attractor for the 2D Navier-Stokes flow on some unbounded domains. Nonlinear Anal, 1998, 32: 71-85

30 Sun C, Cao D, Duan J. Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity. Nonlinearity, 2006, 19: 2645-2665

31 Sun C, Cao D, Duan J. Uniform attractors for nonautonomous wave equations with nonlinear damping. SIAM J Applied Dynamical Systems, 2007, 6: 293-318

32 Temam R. Infinite Dimensional Dynamical Systems in Mechanics and Physics. 2nd Ed. New York: Springer-Verlag, 1997

33 Wang B. Attractors for reaction diffusion equations in unbounded domains. Physica D, 1999, 128: 41-52

34 Wang B. Pullback attractors for the non-autonomous FitzHugh-Nagumo system on unbounded domains. Nonlinear Analysis, TMA, doi:10.1016/j.na.2008.07.011

35 Wang B. Asymptotic behavior of stochastic wave equations with critical exponents on R3. <patent>arXiv:0810.1988v1</patent>, 2008

36 Wang B. Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains. J Differential Equations, 2009, 246: 2506-2537

37 Wang B, Yang W. Finite-dimensional behaviour for the Benjamin-Bona-Mahony equation. J Phys A, 1997, 30: 4877-4885

38 Wang X. An energy equation for the weakly damped driven nonlinear Schrodinger equations and its application to their attractors. Physica D, 1995, 88: 167-175

39 Wang Y,Wang L, Zhao W. Pullback attractors for nonautonomous reaction-diffusion equations in unbounded domains. J Math Anal Appl, 2007, 336: 330-347