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March  2013, 12(2): 1111-1121. doi: 10.3934/cpaa.2013.12.1111

Attractors in $H^2$ and $L^{2p-2}$ for reaction diffusion equations on unbounded domains

1. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China, China

Received  October 2011 Revised  February 2012 Published  September 2012

We are concerned with a class of reaction diffusion equations with nonlinear terms of arbitrary growth on unbounded domains. The existence of an $L^2 - L^{2p-2} \cap H^2$ global attractor is proved. This improves the results in previous references, and the proof is shorter.
Citation: Ming Wang, Yanbin Tang. Attractors in $H^2$ and $L^{2p-2}$ for reaction diffusion equations on unbounded domains. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1111-1121. doi: 10.3934/cpaa.2013.12.1111
References:
[1]

J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. R. Bernal, Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains, Nonlinear Anal., 56 (2004), 515-554. doi: 10.1016/j.na.2003.09.023.

[2]

J. M. Arrieta, N. Moya and A. R. Bernal, On the finite dimension of attractors of parabolic problems in $\mathbbR^N$ with general potentials, Nonlinear Anal., 68 (2008), 1082-1099. doi: 10.1016/j.na.2006.12.007.

[3]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,'' North-Holland, Amsterdam, 1992.

[4]

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

[5]

A. V. Babin and B. Nicolaenko, Exponential attractors of reaction-diffusion systems in an unbounded domain, J. Dyn. Diff. Eqs., 7 (1995), 567-590. doi: 10.1007/BF02218725.

[6]

J. M. Ball, Global Attractors for Damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31.

[7]

J. W. Cholewa and T. Dlotko, Bi-spaces globle attractors in abstract parabolic equations, Evolution equations Banach Center Publications, 60 (2003), 13-26. doi: 10.4064/bc60-0-1.

[8]

D. Daners and S. Merino, Gradient-like parabolic semiflows on BUC on $\mathbbR^n$, Proc. Roy. Soc. Sect. A Edinburgh, 128 (1998), 1281-1291. doi: 10.1017/S0308210500027323.

[9]

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

[10]

L. Grafakos, "Classical and Modern Fourier Analysis,'' Pearson 2nd, 2004.

[11]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,'' American Mathematical Society, Providence, RI, 1988.

[12]

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 Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255.

[13]

S. Merino, On the existence of the compact global attractor for semilinear reaction-diffusion systems on $R^n$, J. Differential Equations, 132 (1996), 87-106. doi: 10.1006/jdeq.1996.0172.

[14]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, Handbook of differential equations, ().  doi: 10.1016/S1874-5717(08)00003-0.

[15]

F. Morillas and J. Valero, Attractors for reaction-diffusion equations in $R^n$ with continuous nonlinearity, Asymptot. Anal., 44 (2005), 111-130.

[16]

M. Prizzi, A remark on reaction-diffusion equations in unbounded domains, Discrete Contin. Dyn. Syst., 9 (2003), 281-286.

[17]

M. Prizzi and K. P. Rybakowski, Attractors for reaction-diffusion equations on arbitrary unbounded domains, Topol. Methods Nonlinear Anal., 30 (2007), 251-277.

[18]

J. C. Robinson, "Infinite-dimensional Dynamical Systems an Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,'' Cambridge University Press, Cambridge, 2001.

[19]

R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Anal. TMA, 32 (1998), 71-85. doi: 10.1016/S0362-546X(97)00453-7.

[20]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,'' Springer, New York, 2002.

[21]

C. Y. Sun and C. K. Zhong, Attractors for the semilinear reaction-diffusion equation with distribution derivatives in unbounded domains, Nonlinear Anal., 63 (2005), 49-65. doi: 10.1016/j.na.2005.04.034.

[22]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics,'' 2nd ed., Springer, New York, 1997.

[23]

V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators, Commun. Pure Appl. Anal., 6 (2007) 481-486. doi: 10.3934/cpaa.2007.6.481.

[24]

B. Wang, Attractors for reaction diffusion equations in unbounded domains, Physica D, (1999) 41 - 52. doi: 10.1016/S0167-2789(98)00304-2.

[25]

Z. Q. Wu, J. X. Yin and C. P. Wang, "Elliptic and Parabolic Equations,'' Singapore: World Scientific, 2006.

[26]

L. Yang and M. H. Yang, Long-time behavior of reaction-diffusion equations with dynamical boundary condition, Nonlinear Anal., 74 (2011), 3876-3883. doi: 10.1016/j.na.2011.02.022.

[27]

M. H. Yang and C. K. Zhong, The existence and uniqueness of the solutions for partly dissipative reaction diffusion systems in $\mathbbR^n$, Journal of Lanzhou University, 3 (2006), 130-136 (in Chinese).

[28]

S. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity, Commun. Pure Appl. Math., 56 (2003), 584-637. doi: 10.1002/cpa.10068.

[29]

Y. H. Zhang and L. G. Mu, Existence of the solution to a class of nonlinear reaction-diffusion equation in $\mathbbR^N$, J. Gansu Normal College, 12 (2007), 1-4 (in Chinese).

[30]

Y. H. Zhang, C. K. Zhong and S. Y. Wang, Attractors in $L^2(\mathbbR^N)$ for a class of reaction-diffusion equations, Nonlinear Anal. TMA, 71 (2009), 1901-1908. doi: 10.1016/j.na.2009.01.025.

[31]

Y. H. Zhang, C. K. Zhong and S. Y. Wang, Attractors in $L^p(\mathbbR^N)$ and $H^1(\mathbbR^N)$ for a class of reaction-diffusion equations, Nonlinear Anal. TMA, 72 (2010), 2228-2237. doi: 10.1016/j.na.2009.10.022.

[32]

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 J. Differential Equations, 223 (2006), 367-399. doi: 10.1016/j.jde.2005.06.008.

show all references

References:
[1]

J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. R. Bernal, Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains, Nonlinear Anal., 56 (2004), 515-554. doi: 10.1016/j.na.2003.09.023.

[2]

J. M. Arrieta, N. Moya and A. R. Bernal, On the finite dimension of attractors of parabolic problems in $\mathbbR^N$ with general potentials, Nonlinear Anal., 68 (2008), 1082-1099. doi: 10.1016/j.na.2006.12.007.

[3]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,'' North-Holland, Amsterdam, 1992.

[4]

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

[5]

A. V. Babin and B. Nicolaenko, Exponential attractors of reaction-diffusion systems in an unbounded domain, J. Dyn. Diff. Eqs., 7 (1995), 567-590. doi: 10.1007/BF02218725.

[6]

J. M. Ball, Global Attractors for Damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31.

[7]

J. W. Cholewa and T. Dlotko, Bi-spaces globle attractors in abstract parabolic equations, Evolution equations Banach Center Publications, 60 (2003), 13-26. doi: 10.4064/bc60-0-1.

[8]

D. Daners and S. Merino, Gradient-like parabolic semiflows on BUC on $\mathbbR^n$, Proc. Roy. Soc. Sect. A Edinburgh, 128 (1998), 1281-1291. doi: 10.1017/S0308210500027323.

[9]

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

[10]

L. Grafakos, "Classical and Modern Fourier Analysis,'' Pearson 2nd, 2004.

[11]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,'' American Mathematical Society, Providence, RI, 1988.

[12]

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 Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255.

[13]

S. Merino, On the existence of the compact global attractor for semilinear reaction-diffusion systems on $R^n$, J. Differential Equations, 132 (1996), 87-106. doi: 10.1006/jdeq.1996.0172.

[14]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, Handbook of differential equations, ().  doi: 10.1016/S1874-5717(08)00003-0.

[15]

F. Morillas and J. Valero, Attractors for reaction-diffusion equations in $R^n$ with continuous nonlinearity, Asymptot. Anal., 44 (2005), 111-130.

[16]

M. Prizzi, A remark on reaction-diffusion equations in unbounded domains, Discrete Contin. Dyn. Syst., 9 (2003), 281-286.

[17]

M. Prizzi and K. P. Rybakowski, Attractors for reaction-diffusion equations on arbitrary unbounded domains, Topol. Methods Nonlinear Anal., 30 (2007), 251-277.

[18]

J. C. Robinson, "Infinite-dimensional Dynamical Systems an Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,'' Cambridge University Press, Cambridge, 2001.

[19]

R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Anal. TMA, 32 (1998), 71-85. doi: 10.1016/S0362-546X(97)00453-7.

[20]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,'' Springer, New York, 2002.

[21]

C. Y. Sun and C. K. Zhong, Attractors for the semilinear reaction-diffusion equation with distribution derivatives in unbounded domains, Nonlinear Anal., 63 (2005), 49-65. doi: 10.1016/j.na.2005.04.034.

[22]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics,'' 2nd ed., Springer, New York, 1997.

[23]

V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators, Commun. Pure Appl. Anal., 6 (2007) 481-486. doi: 10.3934/cpaa.2007.6.481.

[24]

B. Wang, Attractors for reaction diffusion equations in unbounded domains, Physica D, (1999) 41 - 52. doi: 10.1016/S0167-2789(98)00304-2.

[25]

Z. Q. Wu, J. X. Yin and C. P. Wang, "Elliptic and Parabolic Equations,'' Singapore: World Scientific, 2006.

[26]

L. Yang and M. H. Yang, Long-time behavior of reaction-diffusion equations with dynamical boundary condition, Nonlinear Anal., 74 (2011), 3876-3883. doi: 10.1016/j.na.2011.02.022.

[27]

M. H. Yang and C. K. Zhong, The existence and uniqueness of the solutions for partly dissipative reaction diffusion systems in $\mathbbR^n$, Journal of Lanzhou University, 3 (2006), 130-136 (in Chinese).

[28]

S. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity, Commun. Pure Appl. Math., 56 (2003), 584-637. doi: 10.1002/cpa.10068.

[29]

Y. H. Zhang and L. G. Mu, Existence of the solution to a class of nonlinear reaction-diffusion equation in $\mathbbR^N$, J. Gansu Normal College, 12 (2007), 1-4 (in Chinese).

[30]

Y. H. Zhang, C. K. Zhong and S. Y. Wang, Attractors in $L^2(\mathbbR^N)$ for a class of reaction-diffusion equations, Nonlinear Anal. TMA, 71 (2009), 1901-1908. doi: 10.1016/j.na.2009.01.025.

[31]

Y. H. Zhang, C. K. Zhong and S. Y. Wang, Attractors in $L^p(\mathbbR^N)$ and $H^1(\mathbbR^N)$ for a class of reaction-diffusion equations, Nonlinear Anal. TMA, 72 (2010), 2228-2237. doi: 10.1016/j.na.2009.10.022.

[32]

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 J. Differential Equations, 223 (2006), 367-399. doi: 10.1016/j.jde.2005.06.008.

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