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Spatial dynamics of a Lotka-Volterra model with a shifting habitat
Limiting behavior of trajectory attractors of perturbed reaction-diffusion equations
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China |
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.
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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