Abstract
In this chapter we develop the existence theory for pullback attractors in a way that recovers well known results for the global attractors of autonomous systems as a particular case (see, for example, Babin and Vishik 1992; Chepyzhov and Vishik 2002;Cholewa and Dlotko 2000; Chueshov 1999; Hale 1988; Ladyzhenskaya 1991; Robinson 2001; Temam 1988).
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Babin AV, Vishik MI (1992) Attractors of evolution equations. North Holland, Amsterdam
Caraballo T, Langa JA, Valero J (2003) The dimension of attractors of nonautonomous partial differential equations. ANZIAM J 45:207–222
Caraballo T, Marín-Rubio P, Valero J (2005) Autonomous and non-autonomous attractors for differential equations with delays. J Differential Equations 208:9–41
Caraballo T, Lukaszewicz G, Real J (2006a) Pullback attractors for asymptotically compact non-autonomous dynamical systems. Nonlinear Anal 64:484–498
Caraballo T, Carvalho AN, Langa JA, Rivero F (2010a) Existence of pullback attractors for pullback asymptotically compact processes. Nonlinear Anal 72:1967–1976
Carvalho AN, Sonner E (2012) Pullback exponential attractors for evolution processes in Banach spaces: theoretical results, submited
Chafee N, Infante EF (1974) A bifurcation problem for a nonlinear partial differential equation of parabolic type. Appl Anal 4:17–37
Chepyzhov VV, Vishik MI (1994) Attractors of nonautonomous dynamical systems and their dimension. J Math Pures Appl 73:279–333
Chepyzhov VV, Vishik MI (2002) Attractors for equations of mathematical physics. Colloquium Publications 49, American Mathematical Society, Providence, RI
Cholewa J, Dlotko T (2000) Global attractors in abstract parabolic problems. LMS lecture notes series 278. Cambridge University Press, Cambridge
Chueshov ID (1999) Introduction to the theory of infinite-dimensional dissipative systems. University lectures in contemporary mathematics. ACTA, Kharkiv, Ukraine
Crauel H (2001) Random point attractors versus random set attractors. J Lond Math Soc 63: 413–427
Crauel H, Flandoli F (1994) Attractors for random dynamical systems. Probab Theory Related Fields 100:365–393
Crauel H, Debussche A, Flandoli F (1997) Random attractors. J Dynam Differential Equations 9:397–341
Czaja R, Efendiev M (2011) Pullback exponential attractors for nonautonomous equations. I: Semilinear parabolic problems. 381:748–765
Di Plinio F, Duan GS, Temam R (2011) Time-dependent attractor for the Oscillon equation. Discrete Contin Dyn Syst A 29:141–167
Efendiev M, Miranville A, Zelik S (2005) Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems. Proc Roy Soc Edinburgh Sect A 135:703–730
Hale JK (1988) Asymptotic behavior of dissipative systems. Mathematical surveys and monographs, American Mathematival Society, Providence, RI
Kloeden PE, Langa JA (2007) Flattening, squeezing and the existence of random attractors. Proc R Soc Lond Ser A 463:163–181
Kloeden PE, Siegmund S (2005) Bifurcation and continuous transition of attractors in autonomous and nonautonomous systems. Int J Bifur Chaos 15:743–762
Koksch N, Siegmund S (2002) Pullback attracting inertial manifolds for nonautonomous dynamical systems. J Dynam Differential Equations 14:889–941
Ladyzhenskaya OA (1991) Attractors for semigroups and evolution equations. Cambridge University Press, Cambridge
Langa JA, Robinson JC, Suárez A (2002) Stability, instability, and bifurcation phenomena in non-autonomous differential equations. Nonlinearity 15:887–903
Langa JA, Miranville A, Real J (2010a) Pullback exponential attractors. Discrete Contin Dyn Syst 26:1329–1357
Liu Z (2007) The random case of Conley’s theorem. III: Random semiflow case and Morse decomposition. Nonlinearity 20:2773–2791
Lukaszewicz, G (2010) On pullback attractors in H 0 1 for nonautonomous reaction-diffusion equations. Int J Bifurc Chaos 20:2637–2644
Ma QF, Wang SH, Zhong CK (2002) Necessary and sufficient conditions for the existence of global attractors for semigroups and applications. Indiana Univ Math J 51:1541–1559
Rasmussen M (2007b) Morse decompositions of nonautonomous dynamical systems. Trans Amer Math Soc 359:5091–5115
Raugel G (2002) Global attractors in partial differential equations. In: Fiedler B (ed) Handbook of dynamical systems, vol 2. North Holland, Amsterdam, pp. 885–892
Robinson JC (2001) Infinite-dimensional dynamical systems. Cambridge University Press, Cambridge
Temam R (1988) Infinite-dimensional dynamical systems in mechanics and physics. Springer, Berlin Heidelberg New York
Vishik MI (1992) Asymptotic behaviour of solutions of evolutionary equations. Cambridge University Press, Cambridge
Wang B (2009) Pullback attractors for non-autonomous reaction-diffusion equations on ℝ n. Front Math China 4:563–583
Wang Y (2008) Pullback attractors for nonautonomous wave equations with critical exponent. Nonlinear Anal 68:365–376
Wang Y, Zhong C, Zhou S (2006) Pullback attractors of nonautonomous dynamical systems. Discrete Contin Dyn Syst 16:587–614
Wang ZX, Fan XL, Zhu ZY (1998) Inertial manifolds for nonautonomous infinite-dimensional dynamical systems. Appl Math Mech 19:695–704
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Carvalho, A.N., Langa, J.A., Robinson, J.C. (2013). Existence results for pullback attractors. In: Attractors for infinite-dimensional non-autonomous dynamical systems. Applied Mathematical Sciences, vol 182. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4581-4_2
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DOI: https://doi.org/10.1007/978-1-4614-4581-4_2
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