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Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations
Global exponential κ-dissipative semigroups and exponential attraction
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 |
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.
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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.
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Exponential attractors for a nonlinear reaction-diffusion system in R3, Comptes Rendus de l'Académie des Sciences -Series I, 330 (2000), 713-718.
|
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M. Efendiev, A. Miranville and S. Zelik,
Exponential attractors for a singularly perturbed Cahn-Hilliard system, Mathematische Nachrichten, 272 (2004), 11-31.
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M. Efendiev, A. Miranville and S. Zelik,
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doi: 10.1098/rspa.2003.1182. |
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Uniform inertial sets for damped wave equations, Discrete and Continuous Dynamical Systems, 6 (2000), 393-418.
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C. Foias and E. Olson,
Finite fractal dimension and Hölder-Lipschitz parametrization, Indiana University Mathematics Journal, 45 (1996), 603-616.
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C. Foias, G. R. Sell and R. Temam,
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Robust exponential attractors for a family of nonconserved phase-field systems with memory, Discrete and Continuous Dynamical Systems, 12 (2005), 1019-1029.
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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.
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[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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