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March  2021, 20(3): 1263-1296. doi: 10.3934/cpaa.2021020

Gromov-Hausdorff stability of reaction diffusion equations with Robin boundary conditions under perturbations of the domain and equation

1. 

Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea

2. 

Department of Mathematics, Chungnam National University, Daejeon 34134, Korea

* Corresponding author

Received  August 2020 Revised  December 2020 Published  March 2021 Early access  February 2021

Fund Project: The first author is supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF- 2019R1A6A3A01091340) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) No. 2015R1A3A2031159

In this paper, we prove the continuity of global attractors and the Gromov-Hausdorff stability of reaction diffusion equations with Robin boundary conditions under perturbations of the domain and equation if every equilibrium of the unperturbed equation is hyperbolic.

Citation: Jihoon Lee, Nguyen Thanh Nguyen. Gromov-Hausdorff stability of reaction diffusion equations with Robin boundary conditions under perturbations of the domain and equation. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1263-1296. doi: 10.3934/cpaa.2021020
References:
[1]

G. S. AragãoA. L. Pereira and M. C. Pereira, Attractors for a nonlinear parabolic problem with terms concentrating on the boundary, J. Dynam. Differ. Equ., 26 (2014), 871-888.  doi: 10.1007/s10884-014-9412-z.

[2]

A. Arbieto and C. A. Morales, Topological stability from Gromov-Hausdorff viewpoint, Discrete Contin. Dyn. Syst., 37 (2017), 3531-3544.  doi: 10.3934/dcds.2017151.

[3]

J. M. Arrieta and A. N. Carvalho, Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain, J. Differ. Equ., 199 (2004), 143-178.  doi: 10.1016/j.jde.2003.09.004.

[4]

J. M. ArrietaA. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains. III. Continuity of attractors, J. Differ. Equ., 247 (2009), 225-259.  doi: 10.1016/j.jde.2008.12.014.

[5]

J. M. Arrieta, A. N. Carvalho and A. Rodriguez-Bernal, Attractors for parabolic problems with nonlinear boundary condition. Uniform bounds, Commun. Partial Differ. Equ., 25 (2000) 1–37. doi: 10.1080/03605300008821506.

[6]

A. V. Babin and S. Yu. Pilyugin, Continuous dependence of an attractor on the shape of domain, J. Math. Sci., 87 (1997), 3304-3310.  doi: 10.1007/BF02355582.

[7]

P. S. Barbosa and A. L. Pereira, Continuity of attractors for $C^1$ perturbations of a smooth domain, Electron. J. Differ. Equ., 2020 (2020), 1-31. 

[8]

P. S. BarbosaA. L. Pereira and M. C. Pereira, Continuity of attractors for a family of $C^1$ perturbations of the square, Ann. Mat. Pura Appl., 196 (2017), 1365-1398.  doi: 10.1007/s10231-016-0620-5.

[9]

L. A. F. De OliveiraA. L. Pereira and M. C. Pereira, Continuity of attractors for a reaction-diffusion problem with respect to variations of the domain, Electron. J. Differ. Equ., 100 (2005), 1-18. 

[10]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Society for Industrial and Applied Mathematics, Philadelphia, 2011. doi: 10.1137/1.9781611972030.ch1.

[11]

D. B. Henry, Geometric Theory of Semilinear Parabolic Equations, in: Lecture Notes in Mathematics, Springer, Berlin, 1981.

[12] D. B. Henry, Perturbation of the Boundary for Boundary Value Problems, Cambridge Univ. Press, 2005. 
[13]

M. Hurley, Fixed points of topological stable flows, Trans. Amer. Math. Soc., 294 (1986), 625-633.  doi: 10.2307/2000204.

[14]

J. A. LangaJ. C. RobinsonA. Suárez and A. Vidal-López, The stability of attractors for non-autonomous perturbations of gradient-like systems, J. Differ. Equ., 234 (2007), 607-625.  doi: 10.1016/j.jde.2006.11.016.

[15]

J. LeeN. Nguyen and V. M. Toi, Gromov-Hausdorff stability of global attractors of reaction diffusion equations under perturbations of domain, J. Differ. Equ., 269 (2020), 125-147.  doi: 10.1016/j.jde.2019.11.097.

[16]

D. S. Li and P. E. Kloeden, Robustness of asymptotic stability to small time delays, Discrete Contin. Dyn. S., 13 (2005), 1007-1034.  doi: 10.3934/dcds.2005.13.1007.

[17]

A. L. Pereira and M. C. Pereira, Continuity of attractors for a reaction-diffusion problem with nonlinear boundary conditions with respect to variations of the domain, J. Differ. Equ., 239 (2007), 343-370.  doi: 10.1016/j.jde.2007.05.018.

[18] J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, 2001.  doi: 10.1007/978-94-010-0732-0.
[19]

R. F. Thomas, Topological stability: Some fundamental properties, J. Differ. Equ., 59 (1985), 103-122.  doi: 10.1016/0022-0396(85)90140-8.

show all references

References:
[1]

G. S. AragãoA. L. Pereira and M. C. Pereira, Attractors for a nonlinear parabolic problem with terms concentrating on the boundary, J. Dynam. Differ. Equ., 26 (2014), 871-888.  doi: 10.1007/s10884-014-9412-z.

[2]

A. Arbieto and C. A. Morales, Topological stability from Gromov-Hausdorff viewpoint, Discrete Contin. Dyn. Syst., 37 (2017), 3531-3544.  doi: 10.3934/dcds.2017151.

[3]

J. M. Arrieta and A. N. Carvalho, Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain, J. Differ. Equ., 199 (2004), 143-178.  doi: 10.1016/j.jde.2003.09.004.

[4]

J. M. ArrietaA. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains. III. Continuity of attractors, J. Differ. Equ., 247 (2009), 225-259.  doi: 10.1016/j.jde.2008.12.014.

[5]

J. M. Arrieta, A. N. Carvalho and A. Rodriguez-Bernal, Attractors for parabolic problems with nonlinear boundary condition. Uniform bounds, Commun. Partial Differ. Equ., 25 (2000) 1–37. doi: 10.1080/03605300008821506.

[6]

A. V. Babin and S. Yu. Pilyugin, Continuous dependence of an attractor on the shape of domain, J. Math. Sci., 87 (1997), 3304-3310.  doi: 10.1007/BF02355582.

[7]

P. S. Barbosa and A. L. Pereira, Continuity of attractors for $C^1$ perturbations of a smooth domain, Electron. J. Differ. Equ., 2020 (2020), 1-31. 

[8]

P. S. BarbosaA. L. Pereira and M. C. Pereira, Continuity of attractors for a family of $C^1$ perturbations of the square, Ann. Mat. Pura Appl., 196 (2017), 1365-1398.  doi: 10.1007/s10231-016-0620-5.

[9]

L. A. F. De OliveiraA. L. Pereira and M. C. Pereira, Continuity of attractors for a reaction-diffusion problem with respect to variations of the domain, Electron. J. Differ. Equ., 100 (2005), 1-18. 

[10]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Society for Industrial and Applied Mathematics, Philadelphia, 2011. doi: 10.1137/1.9781611972030.ch1.

[11]

D. B. Henry, Geometric Theory of Semilinear Parabolic Equations, in: Lecture Notes in Mathematics, Springer, Berlin, 1981.

[12] D. B. Henry, Perturbation of the Boundary for Boundary Value Problems, Cambridge Univ. Press, 2005. 
[13]

M. Hurley, Fixed points of topological stable flows, Trans. Amer. Math. Soc., 294 (1986), 625-633.  doi: 10.2307/2000204.

[14]

J. A. LangaJ. C. RobinsonA. Suárez and A. Vidal-López, The stability of attractors for non-autonomous perturbations of gradient-like systems, J. Differ. Equ., 234 (2007), 607-625.  doi: 10.1016/j.jde.2006.11.016.

[15]

J. LeeN. Nguyen and V. M. Toi, Gromov-Hausdorff stability of global attractors of reaction diffusion equations under perturbations of domain, J. Differ. Equ., 269 (2020), 125-147.  doi: 10.1016/j.jde.2019.11.097.

[16]

D. S. Li and P. E. Kloeden, Robustness of asymptotic stability to small time delays, Discrete Contin. Dyn. S., 13 (2005), 1007-1034.  doi: 10.3934/dcds.2005.13.1007.

[17]

A. L. Pereira and M. C. Pereira, Continuity of attractors for a reaction-diffusion problem with nonlinear boundary conditions with respect to variations of the domain, J. Differ. Equ., 239 (2007), 343-370.  doi: 10.1016/j.jde.2007.05.018.

[18] J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, 2001.  doi: 10.1007/978-94-010-0732-0.
[19]

R. F. Thomas, Topological stability: Some fundamental properties, J. Differ. Equ., 59 (1985), 103-122.  doi: 10.1016/0022-0396(85)90140-8.

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