We investigate the forwards asymptotic dynamics of non-autonomous differential equations. Our approach is centred on those models for which the vector field is only defined for non-negative times, that is, the laws of evolution are not given, or simply not known, for times before a given time (say time $ t = 0 $). We will be interested in the cases for which the 'driving' (time shift) semigroup has a global attractor in which backwards solutions are not necessarily unique. Considering vector fields in the global attractor of the driving semigroup allows for a natural way to extend vector fields, defined only for non-negative times, to the whole real line. These objects play a crucial role in the description of the asymptotic dynamics of our non-autonomous differential equation. We will study, in some particular cases, the isolated invariant sets of the associated skew-product semigroup with the aim of characterising the global attractor. We develop an example for which we derive decomposition for the global attractor of skew-product semigroup from the characterisation of the attractor of the associated driving semigroup.
Citation: |
[1] |
E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa, Stability of gradient semigroups under perturbations, Nonlinearity, 24 (2011), 2099117.
doi: 10.1088/0951-7715/24/7/010.![]() ![]() ![]() |
[2] |
A. V. Babin and M. I. Vishik, Attractors in Evolutionary Equations, Studies in Mathematics and its Applications 25, North-Holland Publishing Co., Amsterdam, 1992.
![]() ![]() |
[3] |
T. Caraballo, J. C. Jara, J. A. Langa and Z. Liu, Morse decomposition of attractors for non-autonomous dynamical systems, Advanced Nonlinear Studies, to appear.
doi: 10.1515/ans-2013-0204.![]() ![]() ![]() |
[4] |
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.![]() ![]() ![]() |
[5] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences 182, Springer-Verlag, New York, 2013.
doi: 10.1007/978-1-4614-4581-4.![]() ![]() ![]() |
[6] |
D. N. Cheban, Global Attractors of Non-autonomous Dissipative Dynamical Systems, World Scientific, New Jersey, 2004.
doi: 10.1142/9789812563088.![]() ![]() ![]() |
[7] |
V. V. Chepyzhov and M.I. Vishik, Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.
![]() ![]() |
[8] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Colloquium Publications, vol.49. American Mathematical Society, Providence, RI, 2001.
![]() ![]() |
[9] |
C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, 38. American Mathematical Society, Providence, R.I., 1978.
![]() ![]() |
[10] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs Number 25 (American Mathematical Society, Providence, RI), 1988.
![]() ![]() |
[11] |
P. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs 176, AMS, Providence RI, 2011.
doi: 10.1090/surv/176.![]() ![]() ![]() |
[12] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.
doi: 10.1017/CBO9780511569418.![]() ![]() ![]() |
[13] |
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), 483-497.
doi: 10.3934/dcds.2007.18.483.![]() ![]() ![]() |
[14] |
J. A. Langa, J. C. Robinson, A. Suárez and A. Vidal-López, The stability of attractors for non-autonomous perturbations of gradient-like systems, J. Differential Equations, 234 (2007), 607-625.
doi: 10.1016/j.jde.2006.11.016.![]() ![]() ![]() |
[15] |
K. Mischaikow, H. Smith and H. R. Thieme, Asymptotically autonomous semiflows: chain recurrent and Lyapunov functions, Trans. Amer. Math. Soc., 347 (1995), 1669-1685.
doi: 10.2307/2154964.![]() ![]() ![]() |
[16] |
C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lecture Notes in Mathemathics 2002, Springer-Verlag, New York, 2010.
doi: 10.1007/978-3-642-14258-1.![]() ![]() ![]() |
[17] |
M. Rasmussen, Attractivity and Bifurcation for Nonautonomous Dynamical Systems, Lecture Notes in Mathemathics 1907, Springer-Verlag, New York, 2007.
![]() ![]() |
[18] |
K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Universitext, Springer-Verlag, 1987.
doi: 10.1007/978-3-642-72833-4.![]() ![]() ![]() |
[19] |
G. R. Sell, Nonautonomous differential equations and dynamical systems, Trans. Amer. Math. Soc., 127 (1967), 241-283.
doi: 10.2307/1994645.![]() ![]() ![]() |
[20] |
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.![]() ![]() ![]() |
Global attractor for vector feld (2.4) in its positive hull Σ+.