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

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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

Abstract
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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 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.

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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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