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doi: 10.3934/cpaa.2021020

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

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)

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 & Applied Analysis, 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.  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. 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.  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. 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.  Google Scholar

[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.   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. 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.  Google Scholar

[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.  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. 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.  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. 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.  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. 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.  Google Scholar

[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.   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. 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.  Google Scholar

[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.  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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