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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 & 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. doi: 10.1016/j.na.2003.09.023. Google Scholar

[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. doi: 10.1016/j.na.2006.12.007. Google Scholar

[3]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,'', North-Holland, (1992). Google Scholar

[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. doi: 10.1017/S0308210500031498. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[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. doi: 10.1002/cpa.1011. Google Scholar

[10]

L. Grafakos, "Classical and Modern Fourier Analysis,'', Pearson 2nd, (2004). Google Scholar

[11]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,'', American Mathematical Society, (1988). Google Scholar

[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. doi: 10.1512/iumj.2002.51.2255. Google Scholar

[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. doi: 10.1006/jdeq.1996.0172. Google Scholar

[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. Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,'', Springer, (2002). Google Scholar

[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. doi: 10.1016/j.na.2005.04.034. Google Scholar

[22]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics,'', 2nd ed., (1997). Google Scholar

[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. doi: 10.3934/cpaa.2007.6.481. Google Scholar

[24]

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

[25]

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

[26]

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

[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. Google Scholar

[28]

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

[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. Google Scholar

[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. doi: 10.1016/j.na.2009.01.025. Google Scholar

[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. doi: 10.1016/j.na.2009.10.022. Google Scholar

[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. doi: 10.1016/j.jde.2005.06.008. Google Scholar

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. doi: 10.1016/j.na.2003.09.023. Google Scholar

[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. doi: 10.1016/j.na.2006.12.007. Google Scholar

[3]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,'', North-Holland, (1992). Google Scholar

[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. doi: 10.1017/S0308210500031498. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[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. doi: 10.1002/cpa.1011. Google Scholar

[10]

L. Grafakos, "Classical and Modern Fourier Analysis,'', Pearson 2nd, (2004). Google Scholar

[11]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,'', American Mathematical Society, (1988). Google Scholar

[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. doi: 10.1512/iumj.2002.51.2255. Google Scholar

[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. doi: 10.1006/jdeq.1996.0172. Google Scholar

[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. Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,'', Springer, (2002). Google Scholar

[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. doi: 10.1016/j.na.2005.04.034. Google Scholar

[22]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics,'', 2nd ed., (1997). Google Scholar

[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. doi: 10.3934/cpaa.2007.6.481. Google Scholar

[24]

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

[25]

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

[26]

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

[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. Google Scholar

[28]

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

[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. Google Scholar

[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. doi: 10.1016/j.na.2009.01.025. Google Scholar

[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. doi: 10.1016/j.na.2009.10.022. Google Scholar

[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. doi: 10.1016/j.jde.2005.06.008. Google Scholar

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