# American Institute of Mathematical Sciences

October  2019, 24(10): 5673-5694. doi: 10.3934/dcdsb.2019101

## Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations

 Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China

Received  August 2017 Revised  January 2019 Published  June 2019

Fund Project: The author is supported by NSF of China under Grant 11501289.

In this paper, we study the relations between the long-time dynamical behavior of the perturbed reaction-diffusion equations and the exact reaction-diffusion equations with concave and convex nonlinear terms and prove that bounded sets of solutions of the perturbed reaction-diffusion equations converges to the trajectory attractor $\mathscr{U}_0$ of the exact reaction-diffusion equations when $t\rightarrow\infty$ and $\varepsilon\rightarrow 0^+.$ In particular, we show that the trajectory attractor $\mathscr{U}_ \varepsilon$ of the perturbed reaction-diffusion equations converges to the trajectory attractor $\mathscr{U}_0$ of the exact reaction-diffusion equations when $\varepsilon\rightarrow 0^+.$ Moreover, we derive the upper and lower bounds of the fractal dimension for the global attractor of the perturbed reaction-diffusion equations.

Citation: Gaocheng Yue. Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5673-5694. doi: 10.3934/dcdsb.2019101
##### References:
 [1] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Nauka, Moscow, 1989. English trans., North-Holland, Amsterdam, 1992.  Google Scholar [2] A. V. Babin and M. I. Vishik, Attractors of evolutionary partial differential equations and estimates of their dimension, Russian Math. Surveys, 38 (1983), 151-213.   Google Scholar [3] V. V. Chepyzhov, E. S. Titi and M. I. Vishik, On the convergence of solutions of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system, Discrete Contin. Dyn. Syst., 17 (2007), 481-500.  doi: 10.3934/dcds.2007.17.481.  Google Scholar [4] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ. Vol. 49, Amer. Math. Soc. Providence, RI, 2002.  Google Scholar [5] V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for evolution equations, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 1309-1314.  doi: 10.1016/S0021-7824(97)89978-3.  Google Scholar [6] V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl., 76 (1997), 913-964.  doi: 10.1016/S0021-7824(97)89978-3.  Google Scholar [7] V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for reaction-diffusion systems, Topol. Methods Nonlinear Anal., 7 (1996), 49-76.  doi: 10.12775/TMNA.1996.002.  Google Scholar [8] E. C. Crooks, E. N. Dancer and D. Hilhorst, On long-time dynamics for competition-diffusion systems with inhomogeneous Dirichlet boundary conditions, Topol. Methods Nonlinear Anal., 30 (2007), 1-36.   Google Scholar [9] E. Feireisl, Ph. Laurencot and F. Simondon, Global attractors for degenerate parabolic equations on unbounded domain, J. Diff. Equations, 129 (1996), 239-261.  doi: 10.1006/jdeq.1996.0117.  Google Scholar [10] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc. Providence, RJ, 1988.  Google Scholar [11] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, Springer New York, 1981.  Google Scholar [12] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Math. Soc. Providence, 2011. doi: 10.1090/surv/176.  Google Scholar [13] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Leizioni Lincei, Cambridge Univ. Press, Cambridge, New York, 1991.  doi: 10.1017/CBO9780511569418.  Google Scholar [14] J. L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires, Gauthier-Villars, Paris, 1969.  Google Scholar [15] M. Marion, Attractors for reaction-diffusion equations: Existence and estimate of their dimension, Appl. Anal., 25 (1987), 101-147.  doi: 10.1080/00036818708839678.  Google Scholar [16] M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM J. Math. Anal., 20 (1989), 816-844.  doi: 10.1137/0520057.  Google Scholar [17] J. C. Robinson, Infinite-dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001.  doi: 10.1007/978-94-010-0732-0.  Google Scholar [18] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar [19] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-0645-3.  Google Scholar [20] M. I. Vishik, E. S. Titi and V. V. Chepyzhov, On the convergence of trajectory attractors of the three-dimensional Navier-Stokes $\alpha$-model as $\alpha\rightarrow0$.(Russian), Sb. Math., 198 (2007), 1703-1736.  doi: 10.1070/SM2007v198n12ABEH003902.  Google Scholar [21] A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlag, Berlin, Heidelberg, 2010. doi: 10.1007/978-3-642-04631-5.  Google Scholar [22] G. C. Yue, Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces, Discrete Contin. Dyn. Syst. B, 22 (2017), 1645-1671.  doi: 10.3934/dcdsb.2017079.  Google Scholar [23] G. C. Yue and C. K. Zhong, Dynamics of non-autonomous reaction-diffusion equations in locally uniform spaces, Topological Methods in Nonlinear Analysis, 46 (2015), 935-965.   Google Scholar [24] G. C. Yue and C. K. Zhong, Global attractors for the Gray-Scott equations in locally uniform spaces, Discrete Contin. Dyn. Syst. B, 21 (2016), 337-356.  doi: 10.3934/dcdsb.2016.21.337.  Google Scholar [25] 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. Diff. Equations, 223 (2006), 367-399.  doi: 10.1016/j.jde.2005.06.008.  Google Scholar

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##### References:
 [1] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Nauka, Moscow, 1989. English trans., North-Holland, Amsterdam, 1992.  Google Scholar [2] A. V. Babin and M. I. Vishik, Attractors of evolutionary partial differential equations and estimates of their dimension, Russian Math. Surveys, 38 (1983), 151-213.   Google Scholar [3] V. V. Chepyzhov, E. S. Titi and M. I. Vishik, On the convergence of solutions of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system, Discrete Contin. Dyn. Syst., 17 (2007), 481-500.  doi: 10.3934/dcds.2007.17.481.  Google Scholar [4] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Amer. Math. Soc. Colloq. Publ. Vol. 49, Amer. Math. Soc. Providence, RI, 2002.  Google Scholar [5] V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for evolution equations, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 1309-1314.  doi: 10.1016/S0021-7824(97)89978-3.  Google Scholar [6] V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl., 76 (1997), 913-964.  doi: 10.1016/S0021-7824(97)89978-3.  Google Scholar [7] V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for reaction-diffusion systems, Topol. Methods Nonlinear Anal., 7 (1996), 49-76.  doi: 10.12775/TMNA.1996.002.  Google Scholar [8] E. C. Crooks, E. N. Dancer and D. Hilhorst, On long-time dynamics for competition-diffusion systems with inhomogeneous Dirichlet boundary conditions, Topol. Methods Nonlinear Anal., 30 (2007), 1-36.   Google Scholar [9] E. Feireisl, Ph. Laurencot and F. Simondon, Global attractors for degenerate parabolic equations on unbounded domain, J. Diff. Equations, 129 (1996), 239-261.  doi: 10.1006/jdeq.1996.0117.  Google Scholar [10] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc. Providence, RJ, 1988.  Google Scholar [11] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, Springer New York, 1981.  Google Scholar [12] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Math. Soc. Providence, 2011. doi: 10.1090/surv/176.  Google Scholar [13] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Leizioni Lincei, Cambridge Univ. Press, Cambridge, New York, 1991.  doi: 10.1017/CBO9780511569418.  Google Scholar [14] J. L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires, Gauthier-Villars, Paris, 1969.  Google Scholar [15] M. Marion, Attractors for reaction-diffusion equations: Existence and estimate of their dimension, Appl. Anal., 25 (1987), 101-147.  doi: 10.1080/00036818708839678.  Google Scholar [16] M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM J. Math. Anal., 20 (1989), 816-844.  doi: 10.1137/0520057.  Google Scholar [17] J. C. Robinson, Infinite-dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001.  doi: 10.1007/978-94-010-0732-0.  Google Scholar [18] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar [19] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-0645-3.  Google Scholar [20] M. I. Vishik, E. S. Titi and V. V. Chepyzhov, On the convergence of trajectory attractors of the three-dimensional Navier-Stokes $\alpha$-model as $\alpha\rightarrow0$.(Russian), Sb. Math., 198 (2007), 1703-1736.  doi: 10.1070/SM2007v198n12ABEH003902.  Google Scholar [21] A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlag, Berlin, Heidelberg, 2010. doi: 10.1007/978-3-642-04631-5.  Google Scholar [22] G. C. Yue, Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces, Discrete Contin. Dyn. Syst. B, 22 (2017), 1645-1671.  doi: 10.3934/dcdsb.2017079.  Google Scholar [23] G. C. Yue and C. K. Zhong, Dynamics of non-autonomous reaction-diffusion equations in locally uniform spaces, Topological Methods in Nonlinear Analysis, 46 (2015), 935-965.   Google Scholar [24] G. C. Yue and C. K. Zhong, Global attractors for the Gray-Scott equations in locally uniform spaces, Discrete Contin. Dyn. Syst. B, 21 (2016), 337-356.  doi: 10.3934/dcdsb.2016.21.337.  Google Scholar [25] 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. Diff. Equations, 223 (2006), 367-399.  doi: 10.1016/j.jde.2005.06.008.  Google Scholar
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