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June  2017, 37(6): 3487-3502. doi: 10.3934/dcds.2017148

Global exponential κ-dissipative semigroups and exponential attraction

Jin Zhang 1,, , Peter E. Kloeden 2, , Meihua Yang 2, and  Chengkui Zhong 3,

1. 

Department of Mathematics, College of Science, Hohai University, Nanjing 210098, China
2. 

School of Mathematics and Statistics, Huazhong University of Science & Technology, Wuhan 430074, China
3. 

Department of Mathematics, Nanjing University, Nanjing 210093, China

* Corresponding author: Jin Zhang

Received  July 2015 Revised  January 2017 Published  February 2017

Globally exponential $κ-$dissipativity, a new concept of dissipativity for semigroups, is introduced. It provides a more general criterion for the exponential attraction of some evolutionary systems. Assuming that a semigroup $\{S(t)\}_{t≥q 0}$ has a bounded absorbing set, then $\{S(t)\}_{t≥q 0}$ is globally exponentially $κ-$dissipative if and only if there exists a compact set $\mathcal{A}^*$ that is positive invariant and attracts any bounded subset exponentially. The set $\mathcal{A}^*$ need not be finite dimensional. This result is illustrated with an application to a damped semilinear wave equation on a bounded domain.

Keywords: Exponential dissipativity, global attractor, measures of noncompactness.
Mathematics Subject Classification: Primary: 35B41; Secondary: 35K57, 37B55.
Citation: Jin Zhang, Peter E. Kloeden, Meihua Yang, Chengkui Zhong. Global exponential κ-dissipative semigroups and exponential attraction. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3487-3502. doi: 10.3934/dcds.2017148
References:
[1]

A. V. Babin and B. Nicolaenko, Exponential attractors of reaction-diffusion systems in an unbounded domain, Journal of Dynamics and Differential Equations, 7 (1995), 567-590.  doi: 10.1007/BF02218725.

[2]

A. V. Babin and M. I. Vishik, Attractor of Evolution Equations North-Holland Publishing Co. , Amsterdam, 1992.

[3]

I. Chueshov and I. Lasiecka, Long-time behavior of second evolution equations with nonlinear damping Memoirs Amer. Math. Soc. 195 (2008), ⅷ+183 pp. doi: 10.1090/memo/0912.

[4]

P. Constantin, C. Foias, B. Nicolaenko and R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-3506-4.

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K. Deimling, Nonlinear Functional Analysis Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.

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L. Dung and B. Nicolaenko, Exponential attractors in Banach spaces, Journal of Dynamics and Differential Equations, 13 (2001), 791-806.  doi: 10.1023/A:1016676027666.

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A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations John Wiley & Sons, New-York, 1994.

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M. Efendiev and A. Miranville, Finite dimensional attractors for a class of reaction-diffusion equations in Rn with a strong nonlinearity, Discrete and Continuous Dynamical Systems, 5 (1999), 399-424.  doi: 10.3934/dcds.1999.5.399.

[9]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in R3, Comptes Rendus de l'Académie des Sciences -Series I, 330 (2000), 713-718. 

[10]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Mathematische Nachrichten, 272 (2004), 11-31.  doi: 10.1002/mana.200310186.

[11]

M. Efendiev, A. Miranville and S. Zelik, Infinite-dimensional exponential attractors for nonlinear reaction-diffusion systems in unbounded domains and their approximation, Proc. Roy. Soc. London Series, 460 (2004), 1107-1129.  doi: 10.1098/rspa.2003.1182.

[12]

P. Fabrie, C. Galusinski and A. Miranville, Uniform inertial sets for damped wave equations, Discrete and Continuous Dynamical Systems, 6 (2000), 393-418.  doi: 10.3934/dcds.2000.6.393.

[13]

C. Foias and E. Olson, Finite fractal dimension and Hölder-Lipschitz parametrization, Indiana University Mathematics Journal, 45 (1996), 603-616.  doi: 10.1512/iumj.1996.45.1326.

[14]

C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, Journal of Differential Equations, 73 (1988), 309-353.  doi: 10.1016/0022-0396(88)90110-6.

[15]

S. Gatti, M. Grasselli, V. Pata and M. Squassina, Robust exponential attractors for a family of nonconserved phase-field systems with memory, Discrete and Continuous Dynamical Systems, 12 (2005), 1019-1029.  doi: 10.3934/dcds.2005.12.1019.

[16]

J. Hale, Asymptotic Behavior of Dissipative Systems AMS, Providence, RJ, 1988.

[17]

B. R. Hunt and V. Y. Kaloshin, Regularity of embbeddings of infinite-dimensional fractal sets into finite-dimensional spaces, Nonlinearity, 12 (1999), 1263-1275.  doi: 10.1088/0951-7715/12/5/303.

[18]

Q. F. Ma, S. H. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana University Mathematics Journal, 51 (2002), 1541-1559.  doi: 10.1512/iumj.2002.51.2255.

[19]

J. Malek and D. Prazak, Large time behavior by the method of l-trajectories, Journal of Differential Equations, 181 (2002), 243-279.  doi: 10.1006/jdeq.2001.4087.

[20]

R. Mane, On the dimension of the compact invariant sets of certain non-linear maps, dynamical systems and turbulence, Springer Lecture Notes in Mathematics, Springer-Verlag, Heidelberg, 898 (1981), 230-242.

[21]

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

[22]

V. Pata and M. Squassina, On the strongly damped wave equation, Communications in Mathematical Physics, 253 (2005), 511-533.  doi: 10.1007/s00220-004-1233-1.

[23]

D. Prazak, A necessary and sufficient condition for the existence of an exponential attractor, Central European Journal of Mathematics, 1 (2003), 411-417.  doi: 10.2478/BF02475219.

[24]

J. C. Robinson, Infinite-dimensional Dynamical Systems Cambridge University Press, Cambridge, 2002. doi: 10.1007/978-94-010-0732-0.

[25]

J. C. Robinson, Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces, Nonlinearity, 22 (2009), 711-728.  doi: 10.1088/0951-7715/22/4/001.

[26]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.

[27]

M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems 3rd ed. Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-662-02624-3.

[28]

C. Y. Sun, M. H. Yang and C. K. Zhong, Global attractors for the wave equation with nonlinear damping, Journal of Differential Equations, 227 (2006), 427-443.  doi: 10.1016/j.jde.2005.09.010.

[29]

R. Temam, Infinite-Dimensional Systems in Mechanics and Physics Springer-Verlag, New York, 1997.

[30]

S. Zelik, The attractor for a nonlinear reaction-diffusion system in the unbounded domain and Kolmogorov's A-entropy, Mathematische Nachrichten, 232 (2001), 129-179.  doi: 10.1002/1522-2616(200112)232:1<129::AID-MANA129>3.0.CO;2-T.

[31]

C. K. Zhong and W. S. Niu, On the Z2-index of the global attractor for a class of p-Laplacian equations, Nonlinear Analysis, 73 (2010), 3698-3704.  doi: 10.1016/j.na.2010.07.022.

[32]

C. K. Zhong, C. Y. Sun and M. F. Niu, On the existence of global attractor for a class of infinite dimensional dissipative nonlinear dynamical systems, Chinese Annals of Mathematics, Series B, 26 (2005), 393-400.  doi: 10.1142/S0252959905000312.

[33]

C. K. Zhong, M. H. Yang and C. Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, Journal of Differential Equations, 223 (2006), 367-399.  doi: 10.1016/j.jde.2005.06.008.

[34]

Y. S. Zhong and C. K. Zhong, Exponential attractors for semigroups in Banach spaces, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), 1799-1809.  doi: 10.1016/j.na.2011.09.020.

show all references

References:
[1]

A. V. Babin and B. Nicolaenko, Exponential attractors of reaction-diffusion systems in an unbounded domain, Journal of Dynamics and Differential Equations, 7 (1995), 567-590.  doi: 10.1007/BF02218725.

[2]

A. V. Babin and M. I. Vishik, Attractor of Evolution Equations North-Holland Publishing Co. , Amsterdam, 1992.

[3]

I. Chueshov and I. Lasiecka, Long-time behavior of second evolution equations with nonlinear damping Memoirs Amer. Math. Soc. 195 (2008), ⅷ+183 pp. doi: 10.1090/memo/0912.

[4]

P. Constantin, C. Foias, B. Nicolaenko and R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-3506-4.

[5]

K. Deimling, Nonlinear Functional Analysis Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.

[6]

L. Dung and B. Nicolaenko, Exponential attractors in Banach spaces, Journal of Dynamics and Differential Equations, 13 (2001), 791-806.  doi: 10.1023/A:1016676027666.

[7]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations John Wiley & Sons, New-York, 1994.

[8]

M. Efendiev and A. Miranville, Finite dimensional attractors for a class of reaction-diffusion equations in Rn with a strong nonlinearity, Discrete and Continuous Dynamical Systems, 5 (1999), 399-424.  doi: 10.3934/dcds.1999.5.399.

[9]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in R3, Comptes Rendus de l'Académie des Sciences -Series I, 330 (2000), 713-718. 

[10]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Mathematische Nachrichten, 272 (2004), 11-31.  doi: 10.1002/mana.200310186.

[11]

M. Efendiev, A. Miranville and S. Zelik, Infinite-dimensional exponential attractors for nonlinear reaction-diffusion systems in unbounded domains and their approximation, Proc. Roy. Soc. London Series, 460 (2004), 1107-1129.  doi: 10.1098/rspa.2003.1182.

[12]

P. Fabrie, C. Galusinski and A. Miranville, Uniform inertial sets for damped wave equations, Discrete and Continuous Dynamical Systems, 6 (2000), 393-418.  doi: 10.3934/dcds.2000.6.393.

[13]

C. Foias and E. Olson, Finite fractal dimension and Hölder-Lipschitz parametrization, Indiana University Mathematics Journal, 45 (1996), 603-616.  doi: 10.1512/iumj.1996.45.1326.

[14]

C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, Journal of Differential Equations, 73 (1988), 309-353.  doi: 10.1016/0022-0396(88)90110-6.

[15]

S. Gatti, M. Grasselli, V. Pata and M. Squassina, Robust exponential attractors for a family of nonconserved phase-field systems with memory, Discrete and Continuous Dynamical Systems, 12 (2005), 1019-1029.  doi: 10.3934/dcds.2005.12.1019.

[16]

J. Hale, Asymptotic Behavior of Dissipative Systems AMS, Providence, RJ, 1988.

[17]

B. R. Hunt and V. Y. Kaloshin, Regularity of embbeddings of infinite-dimensional fractal sets into finite-dimensional spaces, Nonlinearity, 12 (1999), 1263-1275.  doi: 10.1088/0951-7715/12/5/303.

[18]

Q. F. Ma, S. H. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana University Mathematics Journal, 51 (2002), 1541-1559.  doi: 10.1512/iumj.2002.51.2255.

[19]

J. Malek and D. Prazak, Large time behavior by the method of l-trajectories, Journal of Differential Equations, 181 (2002), 243-279.  doi: 10.1006/jdeq.2001.4087.

[20]

R. Mane, On the dimension of the compact invariant sets of certain non-linear maps, dynamical systems and turbulence, Springer Lecture Notes in Mathematics, Springer-Verlag, Heidelberg, 898 (1981), 230-242.

[21]

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

[22]

V. Pata and M. Squassina, On the strongly damped wave equation, Communications in Mathematical Physics, 253 (2005), 511-533.  doi: 10.1007/s00220-004-1233-1.

[23]

D. Prazak, A necessary and sufficient condition for the existence of an exponential attractor, Central European Journal of Mathematics, 1 (2003), 411-417.  doi: 10.2478/BF02475219.

[24]

J. C. Robinson, Infinite-dimensional Dynamical Systems Cambridge University Press, Cambridge, 2002. doi: 10.1007/978-94-010-0732-0.

[25]

J. C. Robinson, Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces, Nonlinearity, 22 (2009), 711-728.  doi: 10.1088/0951-7715/22/4/001.

[26]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.

[27]

M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems 3rd ed. Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-662-02624-3.

[28]

C. Y. Sun, M. H. Yang and C. K. Zhong, Global attractors for the wave equation with nonlinear damping, Journal of Differential Equations, 227 (2006), 427-443.  doi: 10.1016/j.jde.2005.09.010.

[29]

R. Temam, Infinite-Dimensional Systems in Mechanics and Physics Springer-Verlag, New York, 1997.

[30]

S. Zelik, The attractor for a nonlinear reaction-diffusion system in the unbounded domain and Kolmogorov's A-entropy, Mathematische Nachrichten, 232 (2001), 129-179.  doi: 10.1002/1522-2616(200112)232:1<129::AID-MANA129>3.0.CO;2-T.

[31]

C. K. Zhong and W. S. Niu, On the Z2-index of the global attractor for a class of p-Laplacian equations, Nonlinear Analysis, 73 (2010), 3698-3704.  doi: 10.1016/j.na.2010.07.022.

[32]

C. K. Zhong, C. Y. Sun and M. F. Niu, On the existence of global attractor for a class of infinite dimensional dissipative nonlinear dynamical systems, Chinese Annals of Mathematics, Series B, 26 (2005), 393-400.  doi: 10.1142/S0252959905000312.

[33]

C. K. Zhong, M. H. Yang and C. Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, Journal of Differential Equations, 223 (2006), 367-399.  doi: 10.1016/j.jde.2005.06.008.

[34]

Y. S. Zhong and C. K. Zhong, Exponential attractors for semigroups in Banach spaces, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), 1799-1809.  doi: 10.1016/j.na.2011.09.020.

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