American Institute of Mathematical Sciences

Advanced
  • Previous Article
    The two-component Novikov-type systems with peaked solutions and $ H^1 $-conservation law
  • CPAA Home
  • This Issue
  • Next Article
    Boundary-Domain Integral Equations equivalent to an exterior mixed bvp for the variable-viscosity compressible stokes pdes
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 , and  Nguyen Thanh Nguyen

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.

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, 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

[1]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223
[2]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109
[3]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617
[4]

Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021004
[5]

Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137
[6]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[7]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[8]

Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397
[9]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[10]

Teddy Pichard. A moment closure based on a projection on the boundary of the realizability domain: 1D case. Kinetic & Related Models, 2020, 13 (6) : 1243-1280. doi: 10.3934/krm.2020045
[11]

Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209
[12]

Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637
[13]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175
[14]

Horst R. Thieme. Remarks on resolvent positive operators and their perturbation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 73-90. doi: 10.3934/dcds.1998.4.73
[15]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233
[16]

Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881
[17]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087
[18]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094
[19]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212
[20]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

2019 Impact Factor: 1.105

Tools

Article outline

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences