April  2020, 19(4): 1997-2013. doi: 10.3934/cpaa.2020088

Forwards dynamics of non-autonomous dynamical systems: Driving semigroups without backwards uniqueness and structure of the attractor

1. 

Instituto de Ciências Matemáticas e de Computaçao, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil

2. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Sevilla 41012, Spain

3. 

Mathematical Institute, University of Warwick, Coventry CV4 7AL, UK

*Corresponding author

Dedicated to professor Tomás Caraballo on the occasion of his 60th birthday

Received  August 2019 Revised  October 2019 Published  January 2020

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: Alexandre N. Carvalho, José A. Langa, James C. Robinson. Forwards dynamics of non-autonomous dynamical systems: Driving semigroups without backwards uniqueness and structure of the attractor. Communications on Pure and Applied Analysis, 2020, 19 (4) : 1997-2013. doi: 10.3934/cpaa.2020088
References:
[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. LangaJ. C. RobinsonA. Rodríguez-BernalA. 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. LangaJ. C. RobinsonA. 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. MischaikowH. 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.

show all references

References:
[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. LangaJ. C. RobinsonA. Rodríguez-BernalA. 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. LangaJ. C. RobinsonA. 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. MischaikowH. 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.

Figure 2.2.  Global attractor for vector feld (2.4) in its positive hull Σ+.
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