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Attractors Under Autonomous and Non-autonomous Perturbations
About this Title
Matheus C. Bortolan, Universidade Federal de Santa Catarina, Florianópolis SC, Brazil, Alexandre N. Carvalho, Universidade de São Paulo, São Carlos SP, Brazil and José A. Langa, Universidad de Sevilla, Seville, Spain
Publication: Mathematical Surveys and Monographs
Publication Year:
2020; Volume 246
ISBNs: 978-1-4704-5308-4 (print); 978-1-4704-5684-9 (online)
DOI: https://doi.org/10.1090/surv/246
Table of Contents
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Front/Back Matter
Chapters
Autonomous theory
- Semigroups and global attractors
- Upper and lower semicontinuity
- Topological structural stability of attractors
- Neighborhood of a critical element
- Morse-Smale semigroups
Non-autonomous theory
- Non-autonomous dynamical systems and their attractors
- Upper and lower semicontinuity
- Topological structural stability
- Neighborhood of a global hyperbolic solution
- Non-autonomous Morse-Smale dynamical systems
- S. B. Angenent, The Morse-Smale property for a semilinear parabolic equation, J. Differential Equations 62 (1986), no. 3, 427–442. MR 837763, DOI 10.1016/0022-0396(86)90093-8
- Sigurd Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math. 390 (1988), 79–96. MR 953678, DOI 10.1515/crll.1988.390.79
- Cung The Anh and Nguyen Duong Toan, Existence and upper semicontinuity of uniform attractors in $H^1(\Bbb R^N)$ for nonautonomous nonclassical diffusion equations, Ann. Polon. Math. 111 (2014), no. 3, 271–295. MR 3229425, DOI 10.4064/ap111-3-5
- Eder R. Aragão-Costa, Tomás Caraballo, Alexandre N. Carvalho, and José A. Langa, Continuity of Lyapunov functions and of energy level for a generalized gradient semigroup, Topol. Methods Nonlinear Anal. 39 (2012), no. 1, 57–82. MR 2934334
- E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho, and J. A. Langa, Stability of gradient semigroups under perturbations, Nonlinearity 24 (2011), no. 7, 2099–2117. MR 2805595, DOI 10.1088/0951-7715/24/7/010
- E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho, and J. A. Langa, Non-autonomous Morse-decomposition and Lyapunov functions for gradient-like processes, Trans. Amer. Math. Soc. 365 (2013), no. 10, 5277–5312. MR 3074374, DOI 10.1090/S0002-9947-2013-05810-2
- José M. Arrieta and Alexandre N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations, Trans. Amer. Math. Soc. 352 (2000), no. 1, 285–310. MR 1694278, DOI 10.1090/S0002-9947-99-02528-3
- Jose M. Arrieta and Alexandre N. Carvalho, Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain, J. Differential Equations 199 (2004), no. 1, 143–178. MR 2041515, DOI 10.1016/j.jde.2003.09.004
- José Arrieta, Alexandre N. Carvalho, and Jack K. Hale, A damped hyperbolic equation with critical exponent, Comm. Partial Differential Equations 17 (1992), no. 5-6, 841–866. MR 1177295, DOI 10.1080/03605309208820866
- José M. Arrieta, Alexandre N. Carvalho, José A. Langa, and Aníbal Rodriguez-Bernal, Continuity of dynamical structures for nonautonomous evolution equations under singular perturbations, J. Dynam. Differential Equations 24 (2012), no. 3, 427–481. MR 2964788, DOI 10.1007/s10884-012-9269-y
- José M. Arrieta, Alexandre N. Carvalho, and German Lozada-Cruz, Dynamics in dumbbell domains. I. Continuity of the set of equilibria, J. Differential Equations 231 (2006), no. 2, 551–597. MR 2287897, DOI 10.1016/j.jde.2006.06.002
- José M. Arrieta, Alexandre N. Carvalho, and German Lozada-Cruz, Dynamics in dumbbell domains. II. The limiting problem, J. Differential Equations 247 (2009), no. 1, 174–202. MR 2510133, DOI 10.1016/j.jde.2009.03.014
- José M. Arrieta, Alexandre N. Carvalho, and German Lozada-Cruz, Dynamics in dumbbell domains. III. Continuity of attractors, J. Differential Equations 247 (2009), no. 1, 225–259. MR 2510135, DOI 10.1016/j.jde.2008.12.014
- José M. Arrieta, Alexandre N. Carvalho, Marcone C. Pereira, and Ricardo P. Silva, Semilinear parabolic problems in thin domains with a highly oscillatory boundary, Nonlinear Anal. 74 (2011), no. 15, 5111–5132. MR 2810693, DOI 10.1016/j.na.2011.05.006
- José M. Arrieta, Alexandre N. Carvalho, and Aníbal Rodriguez-Bernal, Critical nonlinearities at the boundary, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), no. 4, 353–358 (English, with English and French summaries). MR 1649955, DOI 10.1016/S0764-4442(99)80047-0
- José M. Arrieta, Alexandre N. Carvalho, and Anibal Rodríguez-Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities, J. Differential Equations 156 (1999), no. 2, 376–406. MR 1705387, DOI 10.1006/jdeq.1998.3612
- José M. Arrieta, Alexandre N. Carvalho, and Aníbal Rodríguez-Bernal, Attractors of parabolic problems with nonlinear boundary conditions. Uniform bounds, Comm. Partial Differential Equations 25 (2000), no. 1-2, 1–37. MR 1737541, DOI 10.1080/03605300008821506
- José M. Arrieta, Alexandre N. Carvalho, and Anibal Rodríguez-Bernal, Upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions, J. Differential Equations 168 (2000), no. 1, 33–59. Special issue in celebration of Jack K. Hale’s 70th birthday, Part 1 (Atlanta, GA/Lisbon, 1998). MR 1801342, DOI 10.1006/jdeq.2000.3876
- José M. Arrieta and Esperanza Santamaría, $C^{1,\theta }$-estimates on the distance of inertial manifolds, Collect. Math. 69 (2018), no. 3, 315–336. MR 3842209, DOI 10.1007/s13348-018-0227-9
- A. V. Babin and M. I. Vishik, Existence and estimates of the dimensions of attractors of quasilinear parabolic equations and of the Navier-Stokes system, Uspekhi Mat. Nauk 37 (1982), no. 3(225), 173–174 (Russian). MR 659432
- A. V. Babin and M. I. Vishik, Regular attractors of semigroups and evolution equations, J. Math. Pures Appl. (9) 62 (1983), no. 4, 441–491 (1984). MR 735932
- —, Attractor in evolutionary equations, vol. 25, Studies in Mathemathics and its Applications, 1992.
- J. M. Ball, On the asymptotic behavior of generalized processes, with applications to nonlinear evolution equations.ations, J. Differential Equations 27 (1978), no. 2, 224–265. MR 461576, DOI 10.1016/0022-0396(78)90032-3
- J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci. 7 (1997), no. 5, 475–502. MR 1462276, DOI 10.1007/s003329900037
- Peter W. Bates, Kening Lu, and Chongchun Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc. 135 (1998), no. 645, viii+129. MR 1445489, DOI 10.1090/memo/0645
- F. D. M. Bezerra, A. N. Carvalho, J. W. Cholewa, and M. J. D. Nascimento, Parabolic approximation of damped wave equations via fractional powers: fast growing nonlinearities and continuity of the dynamics, J. Math. Anal. Appl. 450 (2017), no. 1, 377–405. MR 3606173, DOI 10.1016/j.jmaa.2017.01.024
- S. B. Bodnaruk and V. L. Kulik, On the dependence of bounded invariant manifolds of autonomous systems of differential equations on parameters, Ukraïn. Mat. Zh. 48 (1996), no. 6, 747–752 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 48 (1996), no. 6, 838–845 (1997). MR 1418152, DOI 10.1007/BF02384170
- M. C. Bortolan, T. Caraballo, A. N. Carvalho, and J. A. Langa, An estimate on the fractal dimension of attractors of gradient-like dynamical systems, Nonlinear Anal. 75 (2012), no. 14, 5702–5722. MR 2942948, DOI 10.1016/j.na.2012.05.018
- M. C. Bortolan, T. Caraballo, A. N. Carvalho, and J. A. Langa, Skew product semiflows and Morse decomposition, J. Differential Equations 255 (2013), no. 8, 2436–2462. MR 3082469, DOI 10.1016/j.jde.2013.06.023
- M. C. Bortolan, A. N. Carvalho, and J. A. Langa, Structure of attractors for skew product semiflows, J. Differential Equations 257 (2014), no. 2, 490–522. MR 3200379, DOI 10.1016/j.jde.2014.04.008
- M.C. Bortolan, A.N. Carvalho, J.A. Langa, and Geneviève Raugel, Non-autonomous perturbations of morse-smale semigroups: stability of the phase diagram, Preprint.
- Mahdi Boukrouche, Grzegorz Łukaszewicz, and J. Real, On pullback attractors for a class of two-dimensional turbulent shear flows, Internat. J. Engrg. Sci. 44 (2006), no. 13-14, 830–844. MR 2255762, DOI 10.1016/j.ijengsci.2006.05.012
- R. C. D. S. Broche, A.N. Carvalho, and J. Valero, A non-autonomous scalar one-dimensional dissipative parabolic problem: The description of the dynamics, Preprint.
- P. Brunovský and P. Poláčik, The Morse-Smale structure of a generic reaction-diffusion equation in higher space dimension, J. Differential Equations 135 (1997), no. 1, 129–181. MR 1434918, DOI 10.1006/jdeq.1996.3234
- P. Brunovský and P. Poláčik, The Morse-Smale structure of a generic reaction-diffusion equation in higher space dimension, J. Differential Equations 135 (1997), no. 1, 129–181. MR 1434918, DOI 10.1006/jdeq.1996.3234
- Pavol Brunovsky and Geneviève Raugel, Genericity of the Morse-Smale property for damped wave equations, J. Dynam. Differential Equations 15 (2003), no. 2-3, 571–658. Special issue dedicated to Victor A. Pliss on the occasion of his 70th birthday. MR 2046732, DOI 10.1023/B:JODY.0000009749.10737.9d
- S. M. Bruschi and A. N. Carvalho, Upper semicontinuity of attractors for the discretization of strongly damped wave equations, Mat. Contemp. 32 (2007), 39–62. MR 2428426
- S. M. Bruschi, A. N. Carvalho, J. W. Cholewa, and Tomasz Dlotko, Uniform exponential dichotomy and continuity of attractors for singularly perturbed damped wave equations, J. Dynam. Differential Equations 18 (2006), no. 3, 767–814. MR 2264043, DOI 10.1007/s10884-006-9023-4
- Tomás Caraballo, Alexandre N. Carvalho, Henrique B. da Costa, and José A. Langa, Equi-attraction and continuity of attractors for skew-product semiflows, Discrete Contin. Dyn. Syst. Ser. B 21 (2016), no. 9, 2949–2967. MR 3567795, DOI 10.3934/dcdsb.2016081
- Tomás Caraballo, José A. Langa, Felipe Rivero, and Alexandre N. Carvalho, A gradient-like nonautonomous evolution process, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 20 (2010), no. 9, 2751–2760. MR 2738731, DOI 10.1142/S0218127410027337
- Tomás Caraballo, Alexandre N. Carvalho, José A. Langa, and Felipe Rivero, Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Anal. 72 (2010), no. 3-4, 1967–1976. MR 2577594, DOI 10.1016/j.na.2009.09.037
- Tomás Caraballo, Alexandre N. Carvalho, José A. Langa, and Felipe Rivero, A non-autonomous strongly damped wave equation: existence and continuity of the pullback attractor, Nonlinear Anal. 74 (2011), no. 6, 2272–2283. MR 2781757, DOI 10.1016/j.na.2010.11.032
- Tomás Caraballo and Xiaoying Han, Applied nonautonomous and random dynamical systems, SpringerBriefs in Mathematics, Springer, Cham, 2016. Applied dynamical systems. MR 3586630, DOI 10.1007/978-3-319-49247-6
- Tomás Caraballo, Juan C. Jara, José A. Langa, and Zhenxin Liu, Morse decomposition of attractors for non-autonomous dynamical systems, Adv. Nonlinear Stud. 13 (2013), no. 2, 309–329. MR 3076793, DOI 10.1515/ans-2013-0204
- T. Caraballo, P. E. Kloeden, and P. Marín-Rubio, Weak pullback attractors of setvalued processes, J. Math. Anal. Appl. 288 (2003), no. 2, 692–707. MR 2020190, DOI 10.1016/j.jmaa.2003.09.039
- T. Caraballo, P. E. Kloeden, and J. Real, Pullback and forward attractors for a damped wave equation with delays, Stoch. Dyn. 4 (2004), no. 3, 405–423. MR 2086936, DOI 10.1142/S0219493704001139
- T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 10 (2003), no. 4, 491–513. MR 1978585
- Tomás Caraballo, José A. Langa, and Zhenxin Liu, Gradient infinite-dimensional random dynamical systems, SIAM J. Appl. Dyn. Syst. 11 (2012), no. 4, 1817–1847. MR 3032851, DOI 10.1137/120862752
- T. Caraballo, J. A. Langa, V. S. Melnik, and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal. 11 (2003), no. 2, 153–201. MR 1966698, DOI 10.1023/A:1022902802385
- T. Caraballo, J. A. Langa, and R. Obaya, Pullback, forward and chaotic dynamics in 1D non-autonomous linear-dissipative equations, Nonlinearity 30 (2017), no. 1, 274–299. MR 3604611, DOI 10.1088/1361-6544/30/1/274
- T. Caraballo, G. Łukaszewicz, and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal. 64 (2006), no. 3, 484–498. MR 2191992, DOI 10.1016/j.na.2005.03.111
- T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations 205 (2004), no. 2, 271–297. MR 2091818, DOI 10.1016/j.jde.2004.04.012
- Tomás Caraballo and Stefanie Sonner, Random pullback exponential attractors: general existence results for random dynamical systems in Banach spaces, Discrete Contin. Dyn. Syst. 37 (2017), no. 12, 6383–6403. MR 3690309, DOI 10.3934/dcds.2017277
- Vera Lúcia Carbone, Alexandre N. Carvalho, and Karina Schiabel-Silva, Continuity of attractors for parabolic problems with localized large diffusion, Nonlinear Anal. 68 (2008), no. 3, 515–535. MR 2372362, DOI 10.1016/j.na.2006.11.017
- Vera Lúcia Carbone, Alexandre N. Carvalho, and Karina Schiabel-Silva, Continuity of the dynamics in a localized large diffusion problem with nonlinear boundary conditions, J. Math. Anal. Appl. 356 (2009), no. 1, 69–85. MR 2524216, DOI 10.1016/j.jmaa.2009.02.037
- C. A. Cardoso, J. A. Langa, and R. Obaya, Characterization of cocycle attractors for nonautonomous reaction-diffusion equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 26 (2016), no. 8, 1650135, 20. MR 3533671, DOI 10.1142/S0218127416501352
- Alexandre Nolasco De Carvalho, Spatial homogeneity in damped hyperbolic equations, Dynam. Systems Appl. 1 (1992), no. 3, 221–249. MR 1182646
- Alexandre N. Carvalho, Infinite-dimensional dynamics described by ordinary differential equations, J. Differential Equations 116 (1995), no. 2, 338–404. MR 1318579, DOI 10.1006/jdeq.1995.1039
- A. N. de Carvalho, Parabolic problems with nonlinear boundary conditions in cell tissues, Resenhas 3 (1997), no. 1, 123–138. Workshop on Differential Equations and Nonlinear Analysis (Águas de Lindóia, 1996). MR 1474306
- Alexandre N. Carvalho and Jan W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities, Pacific J. Math. 207 (2002), no. 2, 287–310. MR 1972247, DOI 10.2140/pjm.2002.207.287
- Alexandre N. Carvalho and Jan W. Cholewa, Local well posedness for strongly damped wave equations with critical nonlinearities, Bull. Austral. Math. Soc. 66 (2002), no. 3, 443–463. MR 1939206, DOI 10.1017/S0004972700040296
- A. N. Carvalho and J. W. Cholewa, Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities, J. Math. Anal. Appl. 310 (2005), no. 2, 557–578. MR 2022944, DOI 10.1016/j.jmaa.2005.02.024
- Alexandre N. Carvalho and Jan W. Cholewa, Strongly damped wave equations in $W^{1,p}_0(\Omega )\times L^p(\Omega )$, Discrete Contin. Dyn. Syst. Dynamical systems and differential equations. Proceedings of the 6th AIMS International Conference, suppl. (2007), 230–239. MR 2409217
- A. N. Carvalho and J. W. Cholewa, Regularity of solutions on the global attractor for a semilinear damped wave equation, J. Math. Anal. Appl. 337 (2008), no. 2, 932–948. MR 2386343, DOI 10.1016/j.jmaa.2007.04.051
- Alexandre N. Carvalho and Jan W. Cholewa, Exponential global attractors for semigroups in metric spaces with applications to differential equations, Ergodic Theory Dynam. Systems 31 (2011), no. 6, 1641–1667. MR 2851670, DOI 10.1017/S0143385710000702
- A. N. Carvalho, J. W. Cholewa, and Tomasz Dlotko, Strongly damped wave problems: bootstrapping and regularity of solutions, J. Differential Equations 244 (2008), no. 9, 2310–2333. MR 2413843, DOI 10.1016/j.jde.2008.02.011
- A. N. Carvalho, J. W. Cholewa, and Tomasz Dlotko, Damped wave equations with fast growing dissipative nonlinearities, Discrete Contin. Dyn. Syst. 24 (2009), no. 4, 1147–1165. MR 2505696, DOI 10.3934/dcds.2009.24.1147
- Alexandre N. Carvalho, Jan W. Cholewa, and Tomasz Dłotko, Equi-exponential attraction and rate of convergence of attractors with application to a perturbed damped wave equation, Proc. Roy. Soc. Edinburgh Sect. A 144 (2014), no. 1, 13–51. MR 3164534, DOI 10.1017/S0308210511001235
- A. N. Carvalho, J. W. Cholewa, G. Lozada-Cruz, and M. R. T. Primo, Reduction of infinite dimensional systems to finite dimensions: compact convergence approach, SIAM J. Math. Anal. 45 (2013), no. 2, 600–638. MR 3035474, DOI 10.1137/10080734X
- Alexandre Nolasco de Carvalho and José Alberto Cuminato, Reaction-diffusion problems in cell tissues, J. Dynam. Differential Equations 9 (1997), no. 1, 93–131. MR 1451746, DOI 10.1007/BF02219054
- Alexandre N. Carvalho, Hildebrando M. Rodrigues, and Tomasz Dłotko, Upper semicontinuity of attractors and synchronization, J. Math. Anal. Appl. 220 (1998), no. 1, 13–41. MR 1612059, DOI 10.1006/jmaa.1997.5774
- Alexandre N. Carvalho and Jack K. Hale, Large diffusion with dispersion, Nonlinear Anal. 17 (1991), no. 12, 1139–1151. MR 1137899, DOI 10.1016/0362-546X(91)90233-Q
- Alexandre N. Carvalho and José A. Langa, Non-autonomous perturbation of autonomous semilinear differential equations: continuity of local stable and unstable manifolds, J. Differential Equations 233 (2007), no. 2, 622–653. MR 2292521, DOI 10.1016/j.jde.2006.08.009
- Alexandre N. Carvalho and José A. Langa, An extension of the concept of gradient semigroups which is stable under perturbation, J. Differential Equations 246 (2009), no. 7, 2646–2668. MR 2503016, DOI 10.1016/j.jde.2009.01.007
- Alexandre N. Carvalho, José A. Langa, and James C. Robinson, Lower semicontinuity of attractors for non-autonomous dynamical systems, Ergodic Theory Dynam. Systems 29 (2009), no. 6, 1765–1780. MR 2563091, DOI 10.1017/S0143385708000850
- Alexandre N. Carvalho, José A. Langa, and James C. Robinson, On the continuity of pullback attractors for evolution processes, Nonlinear Anal. 71 (2009), no. 5-6, 1812–1824. MR 2524394, DOI 10.1016/j.na.2009.01.016
- A. N. Carvalho, J. A. Langa, and J. C. Robinson, Structure and bifurcation of pullback attractors in a non-autonomous Chafee-Infante equation, Proc. Amer. Math. Soc. 140 (2012), no. 7, 2357–2373. MR 2898698, DOI 10.1090/S0002-9939-2011-11071-2
- Alexandre N. Carvalho, José A. Langa, and James C. Robinson, Attractors for infinite-dimensional non-autonomous dynamical systems, Applied Mathematical Sciences, vol. 182, Springer, New York, 2013. MR 2976449, DOI 10.1007/978-1-4614-4581-4
- Alexandre N. Carvalho, José A. Langa, and James C. Robinson, Non-autonomous dynamical systems, Discrete Contin. Dyn. Syst. Ser. B 20 (2015), no. 3, 703–747. MR 3331676, DOI 10.3934/dcdsb.2015.20.703
- Alexandre N. Carvalho, José A. Langa, James C. Robinson, and Antonio Suárez, Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system, J. Differential Equations 236 (2007), no. 2, 570–603. MR 2322025, DOI 10.1016/j.jde.2007.01.017
- Alexandre N. Carvalho, José A. Langa, James C. Robinson, and Antonio Suárez, Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system, J. Differential Equations 236 (2007), no. 2, 570–603. MR 2322025, DOI 10.1016/j.jde.2007.01.017
- Alexandre N. Carvalho and Luiz Augusto F. Oliveira, Delay-partial differential equations with some large diffusion, Nonlinear Anal. 22 (1994), no. 9, 1057–1095. MR 1279132, DOI 10.1016/0362-546X(94)90228-3
- Alexandre N. Carvalho and Antônio L. Pereira, A scalar parabolic equation whose asymptotic behavior is dictated by a system of ordinary differential equations, J. Differential Equations 112 (1994), no. 1, 81–130. MR 1287553, DOI 10.1006/jdeq.1994.1096
- Alexandre N. Carvalho and Sergey Piskarev, A general approximation scheme for attractors of abstract parabolic problems, Numer. Funct. Anal. Optim. 27 (2006), no. 7-8, 785–829. MR 2262543, DOI 10.1080/01630560600882723
- Alexandre Nolasco de Carvalho and Marcos Roberto Teixeira Primo, Spatial homogeneity in parabolic problems with nonlinear boundary conditions, Commun. Pure Appl. Anal. 3 (2004), no. 4, 637–651. MR 2106293, DOI 10.3934/cpaa.2004.3.637
- Alexandre N. Carvalho and Stefanie Sonner, Pullback exponential attractors for evolution processes in Banach spaces: theoretical results, Commun. Pure Appl. Anal. 12 (2013), no. 6, 3047–3071. MR 3060923, DOI 10.3934/cpaa.2013.12.3047
- Alexandre N. Carvalho and Stefanie Sonner, Pullback exponential attractors for evolution processes in Banach spaces: properties and applications, Commun. Pure Appl. Anal. 13 (2014), no. 3, 1141–1165. MR 3177693, DOI 10.3934/cpaa.2014.13.1141
- N. Chafee and E. F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type, Applicable Anal. 4 (1974/75), 17–37. MR 440205, DOI 10.1080/00036817408839081
- David N. Cheban, Global attractors of non-autonomous dynamical and control systems, 2nd ed., Interdisciplinary Mathematical Sciences, vol. 18, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. MR 3329680, DOI 10.1142/9297
- D. N. Cheban, P. E. Kloeden, and B. Schmalfuß, The relationship between pullback, forward and global attractors of nonautonomous dynamical systems, Nonlinear Dyn. Syst. Theory 2 (2002), no. 2, 125–144. MR 1989935
- Shu Ping Chen and Roberto Triggiani, Proof of two conjectures by G. Chen and D. L. Russell on structural damping for elastic systems, Approximation and optimization (Havana, 1987) Lecture Notes in Math., vol. 1354, Springer, Berlin, 1988, pp. 234–256. MR 996678, DOI 10.1007/BFb0089601
- Shu Ping Chen and Roberto Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math. 136 (1989), no. 1, 15–55. MR 971932
- Shu Ping Chen and Roberto Triggiani, Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications, J. Differential Equations 88 (1990), no. 2, 279–293. MR 1081250, DOI 10.1016/0022-0396(90)90100-4
- Shu Ping Chen and Roberto Triggiani, Gevrey class semigroups arising from elastic systems with gentle dissipation: the case $0<\alpha <\frac 12$, Proc. Amer. Math. Soc. 110 (1990), no. 2, 401–415. MR 1021208, DOI 10.1090/S0002-9939-1990-1021208-4
- Xu-Yan Chen and Hiroshi Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. Differential Equations 78 (1989), no. 1, 160–190. MR 986159, DOI 10.1016/0022-0396(89)90081-8
- V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl. (9) 73 (1994), no. 3, 279–333. MR 1273705
- Vladimir V. Chepyzhov, Monica Conti, and Vittorino Pata, Totally dissipative dynamical processes and their uniform global attractors, Commun. Pure Appl. Anal. 13 (2014), no. 5, 1989–2004. MR 3230421, DOI 10.3934/cpaa.2014.13.1989
- Vladimir V. Chepyzhov and Mark I. Vishik, Attractors for equations of mathematical physics, American Mathematical Society Colloquium Publications, vol. 49, American Mathematical Society, Providence, RI, 2002. MR 1868930, DOI 10.1051/cocv:2002056
- Alexey Cheskidov and Landon Kavlie, Degenerate pullback attractors for the 3D Navier-Stokes equations, J. Math. Fluid Mech. 17 (2015), no. 3, 411–421. MR 3383921, DOI 10.1007/s00021-015-0214-9
- Alexey Cheskidov and Songsong Lu, Uniform global attractors for the nonautonomous 3D Navier-Stokes equations, Adv. Math. 267 (2014), 277–306. MR 3269180, DOI 10.1016/j.aim.2014.09.005
- Jan W. Cholewa and Tomasz Dlotko, Global attractors in abstract parabolic problems, London Mathematical Society Lecture Note Series, vol. 278, Cambridge University Press, Cambridge, 2000. MR 1778284, DOI 10.1017/CBO9780511526404
- Jan W. Cholewa, Radoslaw Czaja, and Gianluca Mola, Remarks on the fractal dimension of bi-space global and exponential attractors, Boll. Unione Mat. Ital. (9) 1 (2008), no. 1, 121–145 (English, with English and Italian summaries). MR 2388001
- I. D. Chueshov, Vvedenie v teoriyu beskonechnomernykh dissipativnykh sistem, Universitet⋅skie Lektsii po Sovremennoĭ Matematike. [University Lectures in Contemporary Mathematics], AKTA, Kharkiv, 1999 (Russian, with English and Russian summaries). MR 1788405
- Igor Chueshov and Irena Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc. 195 (2008), no. 912, viii+183. MR 2438025, DOI 10.1090/memo/0912
- Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133
- P. Constantin and C. Foias, Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for $2$D Navier-Stokes equations, Comm. Pure Appl. Math. 38 (1985), no. 1, 1–27. MR 768102, DOI 10.1002/cpa.3160380102
- Michele Coti Zelati and Piotr Kalita, Minimality properties of set-valued processes and their pullback attractors, SIAM J. Math. Anal. 47 (2015), no. 2, 1530–1561. MR 3337999, DOI 10.1137/140978995
- Hans Crauel and Franco Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields 100 (1994), no. 3, 365–393. MR 1305587, DOI 10.1007/BF01193705
- Hongyong Cui and José A. Langa, Uniform attractors for non-autonomous random dynamical systems, J. Differential Equations 263 (2017), no. 2, 1225–1268. MR 3632219, DOI 10.1016/j.jde.2017.03.018
- Radoslaw Czaja, Pullback exponential attractors with admissible exponential growth in the past, Nonlinear Anal. 104 (2014), 90–108. MR 3196891, DOI 10.1016/j.na.2014.03.020
- Radosław Czaja, Waldyr M. Oliva, and Carlos Rocha, On a definition of Morse-Smale evolution processes, Discrete Contin. Dyn. Syst. 37 (2017), no. 7, 3601–3623. MR 3639435, DOI 10.3934/dcds.2017155
- Henrique B. da Costa and José Valero, Morse decompositions and Lyapunov functions for dynamically gradient multivalued semiflows, Nonlinear Dynam. 84 (2016), no. 1, 19–34. MR 3472729, DOI 10.1007/s11071-015-2193-z
- E. Brian Davies, Linear operators and their spectra, Cambridge Studies in Advanced Mathematics, vol. 106, Cambridge University Press, Cambridge, 2007. MR 2359869, DOI 10.1017/CBO9780511618864
- Emerson A. M. de Abreu and Alexandre N. Carvalho, Attractors for semilinear parabolic problems with Dirichlet boundary conditions in varying domains, Mat. Contemp. 27 (2004), 37–51. MR 2130211
- A. Eden, C. Foias, B. Nicolaenko, and R. Temam, Exponential attractors for dissipative evolution equations, RAM: Research in Applied Mathematics, vol. 37, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. MR 1335230
- M. Efendiev, S. Zelik, and A. Miranville, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A 135 (2005), no. 4, 703–730. MR 2173336, DOI 10.1017/S030821050000408X
- Pierre Fabrie, Cedric Galusinski, Alain Miranville, and Sergey Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Contin. Dyn. Syst. 10 (2004), no. 1-2, 211–238. Partial differential equations and applications. MR 2026192, DOI 10.3934/dcds.2004.10.211
- Bernold Fiedler and Carlos Rocha, Heteroclinic orbits of semilinear parabolic equations, J. Differential Equations 125 (1996), no. 1, 239–281. MR 1376067, DOI 10.1006/jdeq.1996.0031
- Ciprian Foias, George R. Sell, and Roger Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations 73 (1988), no. 2, 309–353. MR 943945, DOI 10.1016/0022-0396(88)90110-6
- Ciprian Foias and Roger Temam, On the Hausdorff dimension of an attractor for the two-dimensional Navier-Stokes equations, Phys. Lett. A 93 (1983), no. 9, 451–454. MR 697674, DOI 10.1016/0375-9601(83)90628-X
- John M. Franks, Time dependent stable diffeomorphisms, Invent. Math. 24 (1974), 163–172. MR 345136, DOI 10.1007/BF01404304
- Mirelson M. Freitas, Piotr Kalita, and José A. Langa, Continuity of non-autonomous attractors for hyperbolic perturbation of parabolic equations, J. Differential Equations 264 (2018), no. 3, 1886–1945. MR 3721416, DOI 10.1016/j.jde.2017.10.007
- G. Fusco, On the explicit construction of an ODE which has the same dynamics as a scalar parabolic PDE, J. Differential Equations 69 (1987), no. 1, 85–110. MR 897442, DOI 10.1016/0022-0396(87)90104-5
- Giorgio Fusco and Waldyr Muniz Oliva, Jacobi matrices and transversality, Proc. Roy. Soc. Edinburgh Sect. A 109 (1988), no. 3-4, 231–243. MR 963029, DOI 10.1017/S0308210500027748
- G. Fusco and W. M. Oliva, Transversality between invariant manifolds of periodic orbits for a class of monotone dynamical systems, J. Dynam. Differential Equations 2 (1990), no. 1, 1–17. MR 1041196, DOI 10.1007/BF01047768
- F. R. Gantmacher, The theory of matrices. Vols. 1, 2, Chelsea Publishing Co., New York, 1959. Translated by K. A. Hirsch. MR 0107649
- J.-M. Ghidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations, J. Math. Pures Appl. (9) 66 (1987), no. 3, 273–319. MR 913856
- Maurizio Grasselli and Vittorino Pata, Uniform attractors of nonautonomous dynamical systems with memory, Evolution equations, semigroups and functional analysis (Milano, 2000) Progr. Nonlinear Differential Equations Appl., vol. 50, Birkhäuser, Basel, 2002, pp. 155–178. MR 1944162
- G. Guerrero, J. A. Langa, and A. Suárez, Architecture of attractor determines dynamics on mutualistic complex networks, Nonlinear Anal. Real World Appl. 34 (2017), 17–40. MR 3567947, DOI 10.1016/j.nonrwa.2016.07.009
- Jack K. Hale, Ordinary differential equations, 2nd ed., Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980. MR 587488
- J. K. Hale, Asymptotic behaviour and dynamics in infinite dimensions, Nonlinear differential equations (Granada, 1984) Res. Notes in Math., vol. 132, Pitman, Boston, MA, 1985, pp. 1–42. MR 908897
- Jack K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR 941371, DOI 10.1090/surv/025
- Jack K. Hale, Luis T. Magalhães, and Waldyr M. Oliva, Dynamics in infinite dimensions, 2nd ed., Applied Mathematical Sciences, vol. 47, Springer-Verlag, New York, 2002. With an appendix by Krzysztof P. Rybakowski. MR 1914080, DOI 10.1007/b100032
- Jack K. Hale and Geneviève Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Differential Equations 73 (1988), no. 2, 197–214. MR 943939, DOI 10.1016/0022-0396(88)90104-0
- Jack K. Hale and Geneviève Raugel, Lower semicontinuity of attractors of gradient systems and applications, Ann. Mat. Pura Appl. (4) 154 (1989), 281–326. MR 1043076, DOI 10.1007/BF01790353
- Jack K. Hale and Geneviève Raugel, Lower semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Dynam. Differential Equations 2 (1990), no. 1, 19–67. MR 1041197, DOI 10.1007/BF01047769
- Jack K. Hale and Geneviève Raugel, A modified Poincaré method for the persistence of periodic orbits and applications, J. Dynam. Differential Equations 22 (2010), no. 1, 3–68. MR 2607154, DOI 10.1007/s10884-009-9155-4
- A. Haraux, Two remarks on hyperbolic dissipative problems, Nonlinear partial differential equations and their applications. Collège de France seminar, Vol. VII (Paris, 1983–1984) Res. Notes in Math., vol. 122, Pitman, Boston, MA, 1985, pp. 6, 161–179 (English, with French summary). MR 879461
- A. Haraux, Recent results on semi-linear hyperbolic problems in bounded domains, Partial differential equations (Rio de Janeiro, 1986) Lecture Notes in Math., vol. 1324, Springer, Berlin, 1988, pp. 118–126. MR 965530, DOI 10.1007/BFb0100787
- Alain Haraux, Systèmes dynamiques dissipatifs et applications, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 17, Masson, Paris, 1991 (French). MR 1084372
- D.B. Henry, Invariant manifolds near a fixed point, Handwritten notes, IME-USP.
- Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
- Daniel B. Henry, Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations, J. Differential Equations 59 (1985), no. 2, 165–205. MR 804887, DOI 10.1016/0022-0396(85)90153-6
- Dan Henry, Perturbation of the boundary in boundary-value problems of partial differential equations, London Mathematical Society Lecture Note Series, vol. 318, Cambridge University Press, Cambridge, 2005. With editorial assistance from Jack Hale and Antônio Luiz Pereira. MR 2160744, DOI 10.1017/CBO9780511546730
- Luan T. Hoang, Eric J. Olson, and James C. Robinson, On the continuity of global attractors, Proc. Amer. Math. Soc. 143 (2015), no. 10, 4389–4395. MR 3373937, DOI 10.1090/proc/12598
- Mike Hurley, Chain recurrence, semiflows, and gradients, J. Dynam. Differential Equations 7 (1995), no. 3, 437–456. MR 1348735, DOI 10.1007/BF02219371
- Romain Joly and Geneviève Raugel, Generic Morse-Smale property for the parabolic equation on the circle, Ann. Inst. H. Poincaré C Anal. Non Linéaire 27 (2010), no. 6, 1397–1440. MR 2738326, DOI 10.1016/j.anihpc.2010.09.001
- Varga Kalantarov, Anton Savostianov, and Sergey Zelik, Attractors for damped quintic wave equations in bounded domains, Ann. Henri Poincaré 17 (2016), no. 9, 2555–2584. MR 3535872, DOI 10.1007/s00023-016-0480-y
- Peter Kloeden and Thomas Lorenz, Pullback and forward attractors of nonautonomous difference equations, Difference equations, discrete dynamical systems and applications, Springer Proc. Math. Stat., vol. 150, Springer, Cham, 2015, pp. 37–48. MR 3477513, DOI 10.1007/978-3-319-24747-2_{3}
- Peter E. Kloeden, Pullback attractors in nonautonomous difference equations, J. Differ. Equations Appl. 6 (2000), no. 1, 33–52. MR 1752154, DOI 10.1080/10236190008808212
- Peter E. Kloeden and Thomas Lorenz, Pullback attractors of reaction-diffusion inclusions with space-dependent delay, Discrete Contin. Dyn. Syst. Ser. B 22 (2017), no. 5, 1909–1964. MR 3627136, DOI 10.3934/dcdsb.2017114
- Peter Kloeden and Sergey Piskarev, Discrete convergence and the equivalence of equi-attraction and the continuous convergence of attractors, Int. J. Dyn. Syst. Differ. Equ. 1 (2007), no. 1, 38–43. MR 2492244, DOI 10.1504/IJDSDE.2007.013743
- Peter E. Kloeden and Martin Rasmussen, Nonautonomous dynamical systems, Mathematical Surveys and Monographs, vol. 176, American Mathematical Society, Providence, RI, 2011. MR 2808288, DOI 10.1090/surv/176
- Peter E. Kloeden, José Real, and Chunyou Sun, Pullback attractors for a semilinear heat equation on time-varying domains, J. Differential Equations 246 (2009), no. 12, 4702–4730. MR 2523300, DOI 10.1016/j.jde.2008.11.017
- Peter E. Kloeden and Meihua Yang, Forward attraction in nonautonomous difference equations, J. Difference Equ. Appl. 22 (2016), no. 4, 513–525. MR 3486077, DOI 10.1080/10236198.2015.1107550
- H.-W. Knobloch, Invariant manifolds for ordinary differential equations, Differential equations and mathematical physics (Birmingham, AL, 1990) Math. Sci. Engrg., vol. 186, Academic Press, Boston, MA, 1992, pp. 121–149. MR 1126693, DOI 10.1016/S0076-5392(08)63378-0
- Kazuo Kobayasi, An $L^p$ theory of invariant manifolds for parabolic partial differential equations on $\Bbb R^d$, J. Differential Equations 179 (2002), no. 1, 195–212. MR 1883742, DOI 10.1006/jdeq.2001.4026
- S. G. Kryzhevich and V. A. Pliss, On the structural stability of nonautonomous systems, Differ. Uravn. 39 (2003), no. 10, 1325–1333, 1437 (Russian, with Russian summary); English transl., Differ. Equ. 39 (2003), no. 10, 1395–1403. MR 1955030, DOI 10.1023/B:DIEQ.0000017913.79915.b1
- Olga Ladyzhenskaya, Attractors for semigroups and evolution equations, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1991. MR 1133627, DOI 10.1017/CBO9780511569418
- José A. Langa, G. Łukaszewicz, and J. Real, Finite fractal dimension of pullback attractors for non-autonomous 2D Navier-Stokes equations in some unbounded domains, Nonlinear Anal. 66 (2007), no. 3, 735–749. MR 2274880, DOI 10.1016/j.na.2005.12.017
- José A. Langa, Alain Miranville, and José Real, Pullback exponential attractors, Discrete Contin. Dyn. Syst. 26 (2010), no. 4, 1329–1357. MR 2600749, DOI 10.3934/dcds.2010.26.1329
- J. A. Langa, J. C. Robinson, A. Rodríguez-Bernal, A. Suárez, and A. Vidal-López, Existence and nonexistence of unbounded forwards attractor for a class of non-autonomous reaction diffusion equations, Discrete Contin. Dyn. Syst. 18 (2007), no. 2-3, 483–497. MR 2291908, DOI 10.3934/dcds.2007.18.483
- Desheng Li and P. E. Kloeden, Equi-attraction and the continuous dependence of pullback attractors on parameters, Stoch. Dyn. 4 (2004), no. 3, 373–384. MR 2085974, DOI 10.1142/S0219493704001061
- Desheng Li and P. E. Kloeden, Equi-attraction and the continuous dependence of attractors on parameters, Glasg. Math. J. 46 (2004), no. 1, 131–141. MR 2034840, DOI 10.1017/S0017089503001605
- Grzegorz Łukaszewicz and Witold Sadowski, Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains, Z. Angew. Math. Phys. 55 (2004), no. 2, 247–257. MR 2047286, DOI 10.1007/s00033-003-1127-7
- John Mallet-Paret, Negatively invariant sets of compact maps and an extension of a theorem of Cartwright, J. Differential Equations 22 (1976), no. 2, 331–348. MR 423399, DOI 10.1016/0022-0396(76)90032-2
- John Mallet-Paret and George R. Sell, Inertial manifolds for reaction diffusion equations in higher space dimensions, J. Amer. Math. Soc. 1 (1988), no. 4, 805–866. MR 943276, DOI 10.1090/S0894-0347-1988-0943276-7
- Ricardo Mañé, On the dimension of the compact invariant sets of certain nonlinear maps, Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980), Lecture Notes in Math., vol. 898, Springer, Berlin-New York, 1981, pp. 230–242. MR 654892
- Pedro Marín-Rubio and Jose Real, Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators, Discrete Contin. Dyn. Syst. 26 (2010), no. 3, 989–1006. MR 2600726, DOI 10.3934/dcds.2010.26.989
- Hiroshi Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 2, 401–441. MR 672070
- Valery S. Melnik and José Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal. 6 (1998), no. 1, 83–111. MR 1631081, DOI 10.1023/A:1008608431399
- Konstantin Mischaikow, Hal Smith, and Horst R. Thieme, Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions, Trans. Amer. Math. Soc. 347 (1995), no. 5, 1669–1685. MR 1290727, DOI 10.1090/S0002-9947-1995-1290727-7
- M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
- Douglas E. Norton, The fundamental theorem of dynamical systems, Comment. Math. Univ. Carolin. 36 (1995), no. 3, 585–597. MR 1364499
- Waldyr M. Oliva, Morse-Smale semiflows, openness and $A$-stability, Differential equations and dynamical systems (Lisbon, 2000) Fields Inst. Commun., vol. 31, Amer. Math. Soc., Providence, RI, 2002, pp. 285–307. MR 1904521
- W. M. Oliva, J. C. F. De Oliveira, and J. Solà-Morales, An infinite-dimensional Morse-Smale map, NoDEA Nonlinear Differential Equations Appl. 1 (1994), no. 4, 365–387. MR 1300148, DOI 10.1007/BF01194986
- George Osipenko and Eugene Ershov, Perturbation of invariant manifolds of ordinary differential equations, Six lectures on dynamical systems (Augsburg, 1994) World Sci. Publ., River Edge, NJ, 1996, pp. 213–265. MR 1441126
- J. Palis, On Morse-Smale dynamical systems, Topology 8 (1968), 385–404. MR 246316, DOI 10.1016/0040-9383(69)90024-X
- J. Palis and S. Smale, Structural stability theorems, Global Analysis (Proc. Sympos. Pure Math., Vols. XIV, XV, XVI, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 223–231. MR 0267603
- Jacob Palis Jr. and Welington de Melo, Introdução aos sistemas dinâmicos, Projeto Euclides [Euclid Project], vol. 6, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1978 (Portuguese). MR 654863
- Mauro Patrão, Morse decomposition of semiflows on topological spaces, J. Dynam. Differential Equations 19 (2007), no. 1, 181–198. MR 2279951, DOI 10.1007/s10884-006-9033-2
- Mauro Patrão and Luiz A. B. San Martin, Semiflows on topological spaces: chain transitivity and semigroups, J. Dynam. Differential Equations 19 (2007), no. 1, 155–180. MR 2279950, DOI 10.1007/s10884-006-9032-3
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
- P. Poláčik, Parabolic equations: asymptotic behavior and dynamics on invariant manifolds, Handbook of dynamical systems, Vol. 2, North-Holland, Amsterdam, 2002, pp. 835–883. MR 1901067, DOI 10.1016/S1874-575X(02)80037-6
- Martin Rasmussen, All-time Morse decompositions of linear nonautonomous dynamical systems, Proc. Amer. Math. Soc. 136 (2008), no. 3, 1045–1055. MR 2361880, DOI 10.1090/S0002-9939-07-09071-5
- Geneviève Raugel, Singularly perturbed hyperbolic equations revisited, International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999) World Sci. Publ., River Edge, NJ, 2000, pp. 647–652. MR 1870210
- G. Raugel, Global attractors in partial differential equations, Handbook of dynamical systems, Vol. 2, North-Holland, Amsterdam, 2002, pp. 885–982. MR 1901068, DOI 10.1016/S1874-575X(02)80038-8
- Felipe Rivero, Forward and pullback attraction on pullback attractors, Bol. Soc. Esp. Mat. Apl. SeMA 51 (2010), 155–161. MR 2675975, DOI 10.1007/bf03322567
- James C. Robinson, Infinite-dimensional dynamical systems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. An introduction to dissipative parabolic PDEs and the theory of global attractors. MR 1881888, DOI 10.1007/978-94-010-0732-0
- Aníbal Rodríguez-Bernal, Localized spatial homogenization and large diffusion, SIAM J. Math. Anal. 29 (1998), no. 6, 1361–1380. MR 1638046, DOI 10.1137/S003614109731864X
- Aníbal Rodríguez-Bernal and Alejandro Vidal-López, Existence, uniqueness and attractivity properties of positive complete trajectories for non-autonomous reaction-diffusion problems, Discrete Contin. Dyn. Syst. 18 (2007), no. 2-3, 537–567. MR 2291911, DOI 10.3934/dcds.2007.18.537
- Krzysztof P. Rybakowski, The homotopy index and partial differential equations, Universitext, Springer-Verlag, Berlin, 1987. MR 910097, DOI 10.1007/978-3-642-72833-4
- George R. Sell, Nonautonomous differential equations and topological dynamics. I. The basic theory, Trans. Amer. Math. Soc. 127 (1967), 241–262. MR 212313, DOI 10.1090/S0002-9947-1967-0212313-2
- George R. Sell and Yuncheng You, Dynamics of evolutionary equations, Applied Mathematical Sciences, vol. 143, Springer-Verlag, New York, 2002. MR 1873467, DOI 10.1007/978-1-4757-5037-9
- Stephen Smale, Morse inequalities for a dynamical system, Bull. Amer. Math. Soc. 66 (1960), 43–49. MR 117745, DOI 10.1090/S0002-9904-1960-10386-2
- Haitao Song and Hongqing Wu, Pullback attractors of nonautonomous reaction-diffusion equations, J. Math. Anal. Appl. 325 (2007), no. 2, 1200–1215. MR 2270079, DOI 10.1016/j.jmaa.2006.02.041
- Bruno Sportisse, A review of current issues in air pollution modeling and simulation, Comput. Geosci. 11 (2007), no. 2, 159–181. MR 2327965, DOI 10.1007/s10596-006-9036-4
- Chunyou Sun, Daomin Cao, and Jinqiao Duan, Uniform attractors for nonautonomous wave equations with nonlinear damping, SIAM J. Appl. Dyn. Syst. 6 (2007), no. 2, 293–318. MR 2318656, DOI 10.1137/060663805
- Chunyou Sun and Yanbo Yuan, $L^p$-type pullback attractors for a semilinear heat equation on time-varying domains, Proc. Roy. Soc. Edinburgh Sect. A 145 (2015), no. 5, 1029–1052. MR 3406460, DOI 10.1017/S0308210515000177
- Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, 2nd ed., Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997. MR 1441312, DOI 10.1007/978-1-4612-0645-3
- M. I. Vishik, Asymptotic behaviour of solutions of evolutionary equations, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1992. MR 1215488
- M. I. Vishik, S. V. Zelik, and V. V. Chepyzhov, Regular attractors and their nonautonomous perturbations, Mat. Sb. 204 (2013), no. 1, 3–46 (Russian, with Russian summary); English transl., Sb. Math. 204 (2013), no. 1-2, 1–42. MR 3060075, DOI 10.1070/SM2013v204n01ABEH004290
- Bixiang Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations 253 (2012), no. 5, 1544–1583. MR 2927390, DOI 10.1016/j.jde.2012.05.015
- Yonghai Wang, Pengrui Li, and Yuming Qin, Upper semicontinuity of uniform attractors for nonclassical diffusion equations, Bound. Value Probl. , posted on (2017), Paper No. 84, 11. MR 3659996, DOI 10.1186/s13661-017-0816-7
- Yejuan Wang and Shengfan Zhou, Kernel sections and uniform attractors of multi-valued semiprocesses, J. Differential Equations 232 (2007), no. 2, 573–622. MR 2286392, DOI 10.1016/j.jde.2006.07.005
- Sergey Zelik, Strong uniform attractors for non-autonomous dissipative PDEs with non translation-compact external forces, Discrete Contin. Dyn. Syst. Ser. B 20 (2015), no. 3, 781–810. MR 3331678, DOI 10.3934/dcdsb.2015.20.781