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From one-sided dichotomies to two-sided dichotomies
Morse decomposition of global attractors with infinite components
1. | Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain |
2. | Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla |
3. | Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, Avda. de la Universidad, s/n, 03202 Elche |
References:
[1] |
E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa, Stability of gradient semigroups under perturbation, Nonlinearity, 24 (2011), 2099-2117.
doi: 10.1088/0951-7715/24/7/010. |
[2] |
E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa, Continuity of Lyapunov functions and of energy level for a generalized gradient system, Topological Methods Nonl. Anal., 39 (2012), 57-82. |
[3] |
J. M. Arrieta, A. Rodríguez-Bernal and J. Valero, Dynamics of a reaction-diffusion equation with a discontinuous nonlinearity, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 2695-2984.
doi: 10.1142/S0218127406016586. |
[4] |
A. V. Babin and M. I. Vishik, Attractors in Evolutionary Equations, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992. |
[5] |
A. N. Carvalho and J. A. Langa, An extension of the concept of gradient semigroups which is stable under perturbation, J. Differential Equations, 246 (2009), 2646-2668.
doi: 10.1016/j.jde.2009.01.007. |
[6] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Series, 182, Springer, New York, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[7] |
A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Suárez, Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system, J. Differential Equations, 236 (2007), 570-603.
doi: 10.1016/j.jde.2007.01.017. |
[8] |
C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, 38, American Mathematical Society, Providence, R.I., 1978. |
[9] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs Number, 25, American Mathematical Society, Providence, RI, 1988. |
[10] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin, 1981. |
[11] |
M. Hurley, Chain recurrence, semiflows and gradients, J. Dyn. Diff. Equations, 7 (1995), 437-456.
doi: 10.1007/BF02219371. |
[12] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.
doi: 10.1017/CBO9780511569418. |
[13] |
D. E. Norton, The fundamental theorem of dynamical systems, Comment. Math., Univ. Carolinae, 36 (1995), 585-597. |
[14] |
M. Patrao, Morse decomposition of semiflows on topological spaces, J. Dyn. Diff. Equations, 19 (2007), 181-198.
doi: 10.1007/s10884-006-9033-2. |
[15] |
M. Patrao and Luiz A.B. San Martin, Semiflows on topological spaces: Chain transitivity and semigroups, J. Dyn. Diff. Equations, 19 (2007), 155-180.
doi: 10.1007/s10884-006-9032-3. |
[16] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, England, 2001.
doi: 10.1007/978-94-010-0732-0. |
[17] |
K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Universitext, Springer-Verlag, 1987.
doi: 10.1007/978-3-642-72833-4. |
[18] |
G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002.
doi: 10.1007/978-1-4757-5037-9. |
[19] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[20] |
M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations, Cambridge University Press, Cambridge, England, 1992. |
show all references
References:
[1] |
E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa, Stability of gradient semigroups under perturbation, Nonlinearity, 24 (2011), 2099-2117.
doi: 10.1088/0951-7715/24/7/010. |
[2] |
E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa, Continuity of Lyapunov functions and of energy level for a generalized gradient system, Topological Methods Nonl. Anal., 39 (2012), 57-82. |
[3] |
J. M. Arrieta, A. Rodríguez-Bernal and J. Valero, Dynamics of a reaction-diffusion equation with a discontinuous nonlinearity, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 2695-2984.
doi: 10.1142/S0218127406016586. |
[4] |
A. V. Babin and M. I. Vishik, Attractors in Evolutionary Equations, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992. |
[5] |
A. N. Carvalho and J. A. Langa, An extension of the concept of gradient semigroups which is stable under perturbation, J. Differential Equations, 246 (2009), 2646-2668.
doi: 10.1016/j.jde.2009.01.007. |
[6] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Series, 182, Springer, New York, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[7] |
A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Suárez, Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system, J. Differential Equations, 236 (2007), 570-603.
doi: 10.1016/j.jde.2007.01.017. |
[8] |
C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, 38, American Mathematical Society, Providence, R.I., 1978. |
[9] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs Number, 25, American Mathematical Society, Providence, RI, 1988. |
[10] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin, 1981. |
[11] |
M. Hurley, Chain recurrence, semiflows and gradients, J. Dyn. Diff. Equations, 7 (1995), 437-456.
doi: 10.1007/BF02219371. |
[12] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.
doi: 10.1017/CBO9780511569418. |
[13] |
D. E. Norton, The fundamental theorem of dynamical systems, Comment. Math., Univ. Carolinae, 36 (1995), 585-597. |
[14] |
M. Patrao, Morse decomposition of semiflows on topological spaces, J. Dyn. Diff. Equations, 19 (2007), 181-198.
doi: 10.1007/s10884-006-9033-2. |
[15] |
M. Patrao and Luiz A.B. San Martin, Semiflows on topological spaces: Chain transitivity and semigroups, J. Dyn. Diff. Equations, 19 (2007), 155-180.
doi: 10.1007/s10884-006-9032-3. |
[16] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, England, 2001.
doi: 10.1007/978-94-010-0732-0. |
[17] |
K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Universitext, Springer-Verlag, 1987.
doi: 10.1007/978-3-642-72833-4. |
[18] |
G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002.
doi: 10.1007/978-1-4757-5037-9. |
[19] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[20] |
M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations, Cambridge University Press, Cambridge, England, 1992. |
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