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January  2016, 21(1): 337-356. doi: 10.3934/dcdsb.2016.21.337

Global attractors for the Gray-Scott equations in locally uniform spaces

Gaocheng Yue 1, and  Chengkui Zhong 2,

1. 

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China
2. 

Department of Mathematics, Nanjing University, Nanjing 210093

Received  December 2014 Revised  July 2015 Published  November 2015

In this paper, we prove the existence of a $(L_{lu}^2(\mathbb{R}^N)\times L_{lu}^2(\mathbb{R}^N),L_{\rho}^2(\mathbb{R}^N)\times L_{\rho}^2(\mathbb{R}^N))$-global attractor for the solution semigroup generated by the Gray-Scott equations on unbounded domains of space dimension $N\leq3.$
Keywords: asymptotic compactness, Gray-Scott equations, well-posedness, global attractor., locally uniform spaces.
Mathematics Subject Classification: Primary: 37L05, 35B40, 35B41; Secondary: 35K5.
Citation: Gaocheng Yue, Chengkui Zhong. Global attractors for the Gray-Scott equations in locally uniform spaces. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 337-356. doi: 10.3934/dcdsb.2016.21.337
References:
[1]

J. Arrieta, J. Cholewa, T. Dlotko and A. Rodriguez-Bernal, Linear parabolic equations in locally spaces, Math. Models Methods Appl. Sci., 14 (2004), 253-293. doi: 10.1142/S0218202504003234.  Google Scholar

[2]

A. Babin and M. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam, 1992.  Google Scholar

[3]

A. Babin and M. Vishik, Attractors of partial differential evolution equations in an unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 221-243. doi: 10.1017/S0308210500031498.  Google Scholar

[4]

A. Carvalho and T. Dlotko, Partly dissipative systems in locally uniform spaces, Colloq. Math., 100 (2004), 221-242. doi: 10.4064/cm100-2-6.  Google Scholar

[5]

J. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge Univ. Press, Cambridge, 2000. doi: 10.1017/CBO9780511526404.  Google Scholar

[6]

J. Cholewa and T. Dlotko, Bi-spaces global attractors in abstract parabolic equations, Banach Center Pull. Evol. Equ., 60 (2003), 13-26.  Google Scholar

[7]

I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta,Kharkov, 1999, in Russian; English translation: Acta, Kharkov, 2002; see also http://www.emis.de/monographs/Chueshov/.  Google Scholar

[8]

M. Efendiev and S. Zelik, The attractor for a nonlinear reaction-diffusion system in an bounded domain, Comm. Pure Appl. Math., 54 (2001), 625-688. doi: 10.1002/cpa.1011.  Google Scholar

[9]

E. Feireisl, P. Laurencot and F. Simondon, Global attractors for degenerate parabolic equations on unbounded domain, J. Differential Equations, 129 (1996), 239-261. doi: 10.1006/jdeq.1996.0117.  Google Scholar

[10]

E. Feireisl, Bounded, locally compact global attractors for semilinear damped wave equations in $\mathbbR^N$, Differential Integral Equations, 9 (1996), 1147-1156.  Google Scholar

[11]

P. Gray and S. Scott, Chemical Waves and Instabilities, Clarendon, Oxford, 1990. Google Scholar

[12]

J. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988.  Google Scholar

[13]

D. Henry, Geometric Theory of Semilinear Parabolic Problems, Lecture Notes in Mathematics 840, Springer, 1981.  Google Scholar

[14]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740.  Google Scholar

[15]

O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Leizioni Lincei/Canbridge Univ. Press, Cambridge/New York, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar

[16]

Q. Ma, S. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. Journal, 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255.  Google Scholar

[17]

M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM J. Math. Anal., 20 (1989), 816-844. doi: 10.1137/0520057.  Google Scholar

[18]

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

[19]

M. Prizzi and K. Rybakowski, Attractors for Semilinear Damped Wave Equations on Arbitrary Unbounded Domains, Topol. Methods Nonlinear Anal., 31 (2008), 49-82.  Google Scholar

[20]

R. Temam, Infinite-Dimensional Dynamical Systems in Methanics and Physics, second edition, Springer, Berlin, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[21]

H. Xiao, Asypmtotic dynamics of plate equation with a critical exponent on unbounded domain, Nonlinear Anal., 70 (2009), 1288-1301. doi: 10.1016/j.na.2008.02.012.  Google Scholar

[22]

G. Yue and C. Zhong, Global attractors for plate equations with critical exponent in locally uniform spaces, Nonlinear Anal., 71 (2009), 4105-4114. doi: 10.1016/j.na.2009.02.089.  Google Scholar

[23]

G. Yue and C. Zhong, Dynamics of non-autonomous reaction-diffusion equations in locally uniform spaces,, Topological Methods in Nonlinear Analysis, ().   Google Scholar

[24]

B. Wang, Attractors for Reaction-Diffusion equation in unbounded domains, Physica D, 128 (1999), 41-52. doi: 10.1016/S0167-2789(98)00304-2.  Google Scholar

[25]

Y. You, Global attractor of the Gray-Scott equations, Comm. Pure and Appl. Anal., 7 (2008), 947-970. doi: 10.3934/cpaa.2008.7.947.  Google Scholar

[26]

S. Zelik, The attractor for a nonlinear hyperbolic equation in the unbounded domain, Discrete Contin. Dyn. Syst., 7 (2001), 593-641. doi: 10.3934/dcds.2001.7.593.  Google Scholar

[27]

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

show all references

References:
[1]

J. Arrieta, J. Cholewa, T. Dlotko and A. Rodriguez-Bernal, Linear parabolic equations in locally spaces, Math. Models Methods Appl. Sci., 14 (2004), 253-293. doi: 10.1142/S0218202504003234.  Google Scholar

[2]

A. Babin and M. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam, 1992.  Google Scholar

[3]

A. Babin and M. Vishik, Attractors of partial differential evolution equations in an unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 221-243. doi: 10.1017/S0308210500031498.  Google Scholar

[4]

A. Carvalho and T. Dlotko, Partly dissipative systems in locally uniform spaces, Colloq. Math., 100 (2004), 221-242. doi: 10.4064/cm100-2-6.  Google Scholar

[5]

J. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge Univ. Press, Cambridge, 2000. doi: 10.1017/CBO9780511526404.  Google Scholar

[6]

J. Cholewa and T. Dlotko, Bi-spaces global attractors in abstract parabolic equations, Banach Center Pull. Evol. Equ., 60 (2003), 13-26.  Google Scholar

[7]

I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta,Kharkov, 1999, in Russian; English translation: Acta, Kharkov, 2002; see also http://www.emis.de/monographs/Chueshov/.  Google Scholar

[8]

M. Efendiev and S. Zelik, The attractor for a nonlinear reaction-diffusion system in an bounded domain, Comm. Pure Appl. Math., 54 (2001), 625-688. doi: 10.1002/cpa.1011.  Google Scholar

[9]

E. Feireisl, P. Laurencot and F. Simondon, Global attractors for degenerate parabolic equations on unbounded domain, J. Differential Equations, 129 (1996), 239-261. doi: 10.1006/jdeq.1996.0117.  Google Scholar

[10]

E. Feireisl, Bounded, locally compact global attractors for semilinear damped wave equations in $\mathbbR^N$, Differential Integral Equations, 9 (1996), 1147-1156.  Google Scholar

[11]

P. Gray and S. Scott, Chemical Waves and Instabilities, Clarendon, Oxford, 1990. Google Scholar

[12]

J. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988.  Google Scholar

[13]

D. Henry, Geometric Theory of Semilinear Parabolic Problems, Lecture Notes in Mathematics 840, Springer, 1981.  Google Scholar

[14]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740.  Google Scholar

[15]

O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Leizioni Lincei/Canbridge Univ. Press, Cambridge/New York, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar

[16]

Q. Ma, S. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. Journal, 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255.  Google Scholar

[17]

M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM J. Math. Anal., 20 (1989), 816-844. doi: 10.1137/0520057.  Google Scholar

[18]

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

[19]

M. Prizzi and K. Rybakowski, Attractors for Semilinear Damped Wave Equations on Arbitrary Unbounded Domains, Topol. Methods Nonlinear Anal., 31 (2008), 49-82.  Google Scholar

[20]

R. Temam, Infinite-Dimensional Dynamical Systems in Methanics and Physics, second edition, Springer, Berlin, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[21]

H. Xiao, Asypmtotic dynamics of plate equation with a critical exponent on unbounded domain, Nonlinear Anal., 70 (2009), 1288-1301. doi: 10.1016/j.na.2008.02.012.  Google Scholar

[22]

G. Yue and C. Zhong, Global attractors for plate equations with critical exponent in locally uniform spaces, Nonlinear Anal., 71 (2009), 4105-4114. doi: 10.1016/j.na.2009.02.089.  Google Scholar

[23]

G. Yue and C. Zhong, Dynamics of non-autonomous reaction-diffusion equations in locally uniform spaces,, Topological Methods in Nonlinear Analysis, ().   Google Scholar

[24]

B. Wang, Attractors for Reaction-Diffusion equation in unbounded domains, Physica D, 128 (1999), 41-52. doi: 10.1016/S0167-2789(98)00304-2.  Google Scholar

[25]

Y. You, Global attractor of the Gray-Scott equations, Comm. Pure and Appl. Anal., 7 (2008), 947-970. doi: 10.3934/cpaa.2008.7.947.  Google Scholar

[26]

S. Zelik, The attractor for a nonlinear hyperbolic equation in the unbounded domain, Discrete Contin. Dyn. Syst., 7 (2001), 593-641. doi: 10.3934/dcds.2001.7.593.  Google Scholar

[27]

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

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