# American Institute of Mathematical Sciences

• Previous Article
Blow-up for semilinear parabolic equations with critical Sobolev exponent
• CPAA Home
• This Issue
• Next Article
Asymptotic behavior of the solutions of the inhomogeneous Porous Medium Equation with critical vanishing density
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
 [1] Yejuan Wang, Peter E. Kloeden. The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4343-4370. doi: 10.3934/dcds.2014.34.4343 [2] S.V. Zelik. The attractor for a nonlinear hyperbolic equation in the unbounded domain. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 593-641. doi: 10.3934/dcds.2001.7.593 [3] Brahim Alouini. Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1781-1801. doi: 10.3934/cpaa.2015.14.1781 [4] Brahim Alouini, Olivier Goubet. Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 651-677. doi: 10.3934/dcdsb.2014.19.651 [5] Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155 [6] José A. Langa, James C. Robinson, Aníbal Rodríguez-Bernal, A. Suárez, A. Vidal-López. Existence and nonexistence of unbounded forwards attractor for a class of non-autonomous reaction diffusion equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 483-497. doi: 10.3934/dcds.2007.18.483 [7] Pedro Aceves-Sánchez, Christian Schmeiser. Fractional diffusion limit of a linear kinetic equation in a bounded domain. Kinetic & Related Models, 2017, 10 (3) : 541-551. doi: 10.3934/krm.2017021 [8] B. Ambrosio, M. A. Aziz-Alaoui, V. L. E. Phan. Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3787-3797. doi: 10.3934/dcdsb.2018077 [9] Martino Prizzi. A remark on reaction-diffusion equations in unbounded domains. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 281-286. doi: 10.3934/dcds.2003.9.281 [10] Kazuhisa Ichikawa, Mahemauti Rouzimaimaiti, Takashi Suzuki. Reaction diffusion equation with non-local term arises as a mean field limit of the master equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 115-126. doi: 10.3934/dcdss.2012.5.115 [11] Emmanuel Hebey and Frederic Robert. Compactness and global estimates for the geometric Paneitz equation in high dimensions. Electronic Research Announcements, 2004, 10: 135-141. [12] Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493 [13] María del Mar González, Regis Monneau. Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1255-1286. doi: 10.3934/dcds.2012.32.1255 [14] Dieter Bothe, Michel Pierre. The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 49-59. doi: 10.3934/dcdss.2012.5.49 [15] Dalibor Pražák. Exponential attractor for the delayed logistic equation with a nonlinear diffusion. Conference Publications, 2003, 2003 (Special) : 717-726. doi: 10.3934/proc.2003.2003.717 [16] Boris Andreianov, Halima Labani. Preconditioning operators and $L^\infty$ attractor for a class of reaction-diffusion systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2179-2199. doi: 10.3934/cpaa.2012.11.2179 [17] Carlos Arnoldo Morales, M. J. Pacifico. Lyapunov stability of $\omega$-limit sets. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 671-674. doi: 10.3934/dcds.2002.8.671 [18] Liangwei Wang, Jingxue Yin, Chunhua Jin. $\omega$-limit sets for porous medium equation with initial data in some weighted spaces. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 223-236. doi: 10.3934/dcdsb.2013.18.223 [19] Wei Wang, Anthony Roberts. Macroscopic discrete modelling of stochastic reaction-diffusion equations on a periodic domain. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 253-273. doi: 10.3934/dcds.2011.31.253 [20] Peter Poláčik, Eiji Yanagida. Stable subharmonic solutions of reaction-diffusion equations on an arbitrary domain. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 209-218. doi: 10.3934/dcds.2002.8.209

2018 Impact Factor: 0.925