September  2019, 8(3): 663-668. doi: 10.3934/eect.2019031

Two questions arising in the theory of attractors

Politecnico di Milano - Dipartimento di Matematica, Via Bonardi 9, 20133 Milano, Italy

Received  February 2019 Revised  May 2019 Published  September 2019 Early access  May 2019

In this note, we dwell on the notions of global and exponential attractors for strongly continuous semigroups acting on a complete metric space. Two natural questions arising in the theory are addressed.

Citation: Vittorino Pata. Two questions arising in the theory of attractors. Evolution Equations and Control Theory, 2019, 8 (3) : 663-668. doi: 10.3934/eect.2019031
References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-HollandAmsterdam, 1992.

[2]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Providence, 2002.

[3]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, MassonParis, 1994.

[4]

M. EfendievA. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in ${\mathbb R}^3$, C.R. Acad. Sci. Paris Sér. I Math., 330 (2000), 713-718.  doi: 10.1016/S0764-4442(00)00259-7.

[5]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc. Providence, 1988.

[6]

A. Haraux, Systèmes Dynamiques Dissipatifs Et Applications, MassonParis, 1991.

[7]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of Differential Equations: Evolutionary Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 4 (2008), 103–200. doi: 10.1016/S1874-5717(08)00003-0.

[8]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[9]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-HollandAmsterdam, 1992.

[2]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Providence, 2002.

[3]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, MassonParis, 1994.

[4]

M. EfendievA. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in ${\mathbb R}^3$, C.R. Acad. Sci. Paris Sér. I Math., 330 (2000), 713-718.  doi: 10.1016/S0764-4442(00)00259-7.

[5]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc. Providence, 1988.

[6]

A. Haraux, Systèmes Dynamiques Dissipatifs Et Applications, MassonParis, 1991.

[7]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of Differential Equations: Evolutionary Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 4 (2008), 103–200. doi: 10.1016/S1874-5717(08)00003-0.

[8]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[9]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

Figure 1.  Trajectories of the semigroup S(t)
Figure 2.  Portrait of the exponential attractor ε
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