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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

Jihoon Lee 1,, and  Nguyen Thanh Nguyen 2,

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  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.

Keywords: Gromov-Haudorff distance, reaction diffusion equations, Robin boundary condition, perturbation of the domain and equation, global attractors.
Mathematics Subject Classification: Primary: 35B40, 35B41, 35K20; Secondary: 58D25.
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 & Applied Analysis, 2021, 20 (3) : 1263-1296. doi: 10.3934/cpaa.2021020
References:
[1]

G. S. Aragão, A. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. M. Arrieta, A. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[8]

P. S. Barbosa, A. 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.  Google Scholar

[9]

L. A. F. De Oliveira, A. 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.   Google Scholar

[10]

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

[11]

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

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

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

[14]

J. A. Langa, J. C. Robinson, A. 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.  Google Scholar

[15]

J. Lee, N. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

show all references

References:
[1]

G. S. Aragão, A. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. M. Arrieta, A. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[8]

P. S. Barbosa, A. 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.  Google Scholar

[9]

L. A. F. De Oliveira, A. 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.   Google Scholar

[10]

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

[11]

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

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

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

[14]

J. A. Langa, J. C. Robinson, A. 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.  Google Scholar

[15]

J. Lee, N. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

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