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

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

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 $N\leq3.$
Citation: Gaocheng Yue, Chengkui Zhong. Global attractors for the Gray-Scott equations in locally uniform spaces. Discrete and 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.

[2]

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

[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.

[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.

[5]

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

[6]

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

[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/.

[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.

[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.

[10]

E. Feireisl, Bounded, locally compact global attractors for semilinear damped wave equations in $\mathbb{R}^N2$, Differential Integral Equations, 9 (1996), 1147-1156.

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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.

[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.

[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.

[19]

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

[20]

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

[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.

[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.

[23]

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

[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.

[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.

[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.

[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.

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.

[2]

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

[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.

[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.

[5]

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

[6]

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

[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/.

[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.

[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.

[10]

E. Feireisl, Bounded, locally compact global attractors for semilinear damped wave equations in $\mathbb{R}^N2$, Differential Integral Equations, 9 (1996), 1147-1156.

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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.

[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.

[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.

[19]

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

[20]

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

[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.

[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.

[23]

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

[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.

[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.

[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.

[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.

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