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Dynamics of harmful algae with seasonal temperature variations in the cove-main lake
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 |
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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