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

Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations

Gaocheng Yue

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

Received  August 2017 Revised  January 2019 Published  October 2019 Early access  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.

Keywords: Reaction-diffusion equations, perturbation, trajectory attractor, global attractor, convergence, fractal dimension.
Mathematics Subject Classification: Primary: 35K57, 37L30, 35B40; Secondary: 35B25.
Citation: Gaocheng Yue. Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations. Discrete and 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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[10]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc. Providence, RJ, 1988.

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, Springer New York, 1981.

[12]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Math. Soc. Providence, 2011. doi: 10.1090/surv/176.

[13] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Leizioni Lincei, Cambridge Univ. Press, Cambridge, New York, 1991.  doi: 10.1017/CBO9780511569418.
[14]

J. L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires, Gauthier-Villars, Paris, 1969.

[15]

M. Marion, Attractors for reaction-diffusion equations: Existence and estimate of their dimension, Appl. Anal., 25 (1987), 101-147.  doi: 10.1080/00036818708839678.

[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.

[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.
[18]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2002. doi: 10.1007/978-1-4757-5037-9.

[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.

[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.

[21]

A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlag, Berlin, Heidelberg, 2010. doi: 10.1007/978-3-642-04631-5.

[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.

[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. 

[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.

[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.

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Nauka, Moscow, 1989. English trans., North-Holland, Amsterdam, 1992.

[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. 

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[10]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc. Providence, RJ, 1988.

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, Springer New York, 1981.

[12]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Math. Soc. Providence, 2011. doi: 10.1090/surv/176.

[13] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Leizioni Lincei, Cambridge Univ. Press, Cambridge, New York, 1991.  doi: 10.1017/CBO9780511569418.
[14]

J. L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires, Gauthier-Villars, Paris, 1969.

[15]

M. Marion, Attractors for reaction-diffusion equations: Existence and estimate of their dimension, Appl. Anal., 25 (1987), 101-147.  doi: 10.1080/00036818708839678.

[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.

[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.
[18]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2002. doi: 10.1007/978-1-4757-5037-9.

[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.

[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.

[21]

A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlag, Berlin, Heidelberg, 2010. doi: 10.1007/978-3-642-04631-5.

[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.

[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. 

[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.

[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.

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