doi: 10.3934/cpaa.2021040

Homogenization of a modified bidomain model involving imperfect transmission

1. 

Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza - Università di Roma, Via A. Scarpa 16, 00161 Roma, Italy

2. 

University of Bucharest, Faculty of Physics, 405, Atomistilor, 077125 Bucharest-Magurele, Romania

* Corresponding author

Received  September 2020 Revised  January 2021 Published  March 2021

Fund Project: The first author is member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The second author is member of the Gruppo Nazionale per la Fisica Matematica (GNFM) of the Istituto Nazionale di Alta Matematica (INdAM)

We study, by means of the periodic unfolding technique, the homogenization of a modified bidomain model, which describes the propagation of the action potential in the cardiac electrophysiology. Such a model, allowing the presence of pathological zones in the heart, involves various geometries and non-standard transmission conditions on the interface between the healthy and the damaged part of the cardiac muscle.

Citation: Micol Amar, Daniele Andreucci, Claudia Timofte. Homogenization of a modified bidomain model involving imperfect transmission. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021040
References:
[1]

E. AcerbiV. Chiadò PiatG. Dal Maso and D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Anal., 18 (1992), 481-496.  doi: 10.1016/0362-546X(92)90015-7.  Google Scholar

[2]

M. AmarD. Andreucci and D. Bellaveglia, Homogenization of an alternating Robin-Neumann boundary condition via time-periodic unfolding, Nonlinear Anal. Theory Methods Appl., 153 (2017), 56-77.  doi: 10.1016/j.na.2016.05.018.  Google Scholar

[3]

M. AmarD. Andreucci and D. Bellaveglia, The time-periodic unfolding operator and applications to parabolic homogenization, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 28 (2017), 663-700.  doi: 10.4171/RLM/781.  Google Scholar

[4]

M. AmarD. AndreucciP. Bisegna and R. Gianni, Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues, Math. Models Methods Appl. Sci, 14 (2004), 1261-1295.  doi: 10.1142/S0218202504003623.  Google Scholar

[5]

M. AmarD. AndreucciP. Bisegna and R. Gianni, Existence and uniqueness for an elliptic problem with evolution arising in electrodynamics, Nonlinear Anal. Real World Appl., 6 (2005), 367-380.  doi: 10.1016/j.nonrwa.2004.09.002.  Google Scholar

[6]

M. AmarD. AndreucciP. Bisegna and R. Gianni, On a hierarchy of models for electrical conduction in biological tissues, Math. Methods Appl. Sci., 29 (2006), 767-787.  doi: 10.1002/mma.709.  Google Scholar

[7]

M. AmarD. AndreucciP. Bisegna and R. Gianni, A hierarchy of models for the electrical conduction in biological tissues via two-scale convergence: The nonlinear case, Differ. Integral Equ., 26 (2013), 885-912.   Google Scholar

[8]

M. Amar, D. Andreucci, R. Gianni and C. Timofte, Concentration and homogenization in electrical conduction in heterogeneous media involving the Laplace-Beltrami operator, Calc. Var., 59: 99 (2020). doi: 10.1007/s00526-020-01749-x.  Google Scholar

[9]

M. Amar, D. Andreucci and C. Timofte, Well-posedness for a modified bidomain model describing bioelectric activity in damaged heart tissue, preprint, arXiv: 2101.09285. Google Scholar

[10]

M. Amar, I. De Bonis and G. Riey, Homogenization of elliptic problems involving interfaces and singular data, Nonlinear Anal., 189 (2019), 111562. Corrigendum to Homogenization of elliptic problems involving interfaces and singular data. Nonlinear Analysis 203 (2021), 112192. doi: 10.1016/j.na.2020.112192.  Google Scholar

[11]

M. Amar and R. Gianni, Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices, Discrete Contin. Dyn. Systems - B, (4)23 (2018), 1739-1756.  doi: 10.3934/dcdsb.2018078.  Google Scholar

[12]

M. Bendahmane and H. Karlsen, Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue, Netw. Heterog. Media, 1 (2006), 185-218.  doi: 10.3934/nhm.2006.1.185.  Google Scholar

[13]

M. Boulakia, Etude mathématique et numérique de modèles issus du domaine biomédical, Equations aux dérivées partielles, UPMC, 2015. Google Scholar

[14]

M. BoulakiaS. CazeauM. A. FernándezJ. F. Gerbeau and N. Zemzemi, Mathematical modeling of electrocardiograms: a numerical study, Ann. Biomed. Eng., 38 (2010), 1071-1097.   Google Scholar

[15]

M. Boulakia, M. A. Fernández, J. F. Gerbeau and N. Zemzemi, Towards the numerical simulation of electrocardiograms, in Functional Imaging and Modeling of the Heart. FIMH 2007. In Lecture Notes in Computer Science (eds. F. Sachse and G. Seemann), Springer, Berlin, 2007, 240–249. Google Scholar

[16]

Y. BourgaultY. Coudière and C. Pierre, Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology, Nonlinear Anal. Real World Appl., (1) (2009), 458-482.  doi: 10.1016/j.nonrwa.2007.10.007.  Google Scholar

[17]

D. CioranescuA. DamlamianP. DonatoG. Griso and R. Zaki, The periodic unfolding method in domains with holes, SIAM J. Math Anal., 44 (2012), 718-760.  doi: 10.1137/100817942.  Google Scholar

[18]

D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method. Theory and Applications to Partial Differential Problems, Springer, Singapore, 2018. doi: 10.1007/978-981-13-3032-2.  Google Scholar

[19]

D. Cioranescu and J. S. J. Paulin, Homogenization in open sets with holes, J. Math. Anal. Appl., 71 (1979), 590-607.  doi: 10.1016/0022-247X(79)90211-7.  Google Scholar

[20]

A. Collin and S. Imperiale, Mathematical analysis and $2$-scale convergence of an heterogeneous microscopic bidomain model, Math. Models Meth. Appl. Sci., 28 (2018), 979-1035.  doi: 10.1142/S0218202518500264.  Google Scholar

[21]

Y. Coudière, A. Davidovic and C. Poignard, Modified bidomain model with passive periodic heterogeneities, Discrete Contin. Dyn. Systems-S, 13 (2020), 2231-2258. doi: 10.3934/dcdss.2020126.  Google Scholar

[22]

A. Davidovi$\grave{\rm c}$, Multiscale Mathematical Modelling of Structural Heterogeneities in Cardiac Electrophysiology, General Mathematics, Universitè de Bordeaux, 2016. Google Scholar

[23]

P. Donato and K. Le Nguyen, Homogenization for diffusion problems with a nonlinear interfacial resistance, Nonlinear Differ. Equ. Appl., 22 (2015), 1345-1380.  doi: 10.1007/s00030-015-0325-2.  Google Scholar

[24]

A. Gaudiello and M. Lenczner, A two-dimensional electrostatic model of interdigitated comb drive in longitudinal mode, Siam J. Appl. Math., 80 (2020), 792-813.  doi: 10.1137/19M1270306.  Google Scholar

[25]

P. GoelJ. Sneyd and A. Friedman, Homogenization of the cell cytoplasm: The calcium bidomain equations, Multiscale Model. Simul., 5 (2006), 1045-1062.  doi: 10.1137/060660783.  Google Scholar

[26]

I. Graf and M. Peter, Diffusion on surfaces and the boundary periodic unfolding operator with an application to carcinogenesis in human cells, SIAM J. Math. Anal., 46 (2014), 3025-3049.  doi: 10.1137/130921015.  Google Scholar

[27]

I. GrafM. Peter and J. Sneyd, Homogenization of a nonlinear multiscale model of calcium dynamics in biological cells, J. Math. Anal. Appl., 419 (2014), 28-47.  doi: 10.1016/j.jmaa.2014.04.037.  Google Scholar

[28]

E. Grandelius and K. Karlsen, The cardiac bidomain model and homogenization, Netw. Heterog. Media, 14 (2019), 173-204.  doi: 10.3934/nhm.2019009.  Google Scholar

[29]

E. HigginsP. GoelJ. PuglisiD. BersM. Cannell and J. Sneyd, Modelling calcium microdomains using homogenisation, J. Theor. Biol., 247 (2007), 623-644.  doi: 10.1016/j.jtbi.2007.03.019.  Google Scholar

[30]

M. Höpker, Extension operators for Sobolev spaces on periodic domains, their applications, and homogenization of a phase field model for phase transitions in porous media, Ph. D. Thesis, Universit$\ddot{a}$t Bremen, 2016.  Google Scholar

[31]

C. Jerez-HanckesI. Pettersson and V. Rybalko, Derivation of cable equation by multiscale analysis for a model of myelinated axons, Discrete Contin. Dyn. Systems-B, 25 (2020), 815-839.  doi: 10.3934/dcdsb.2019191.  Google Scholar

[32]

N. Kajiwara, On the bidomain equations as parabolic evolution equations, Preprint, available from https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/251618/1/2090-02.pdf. Google Scholar

[33]

J. Keener and J. Sneyd, Mathematical Physiology, Springer, 2004.  Google Scholar

[34]

W. Krassowska and J. Neu, Homogenization of syncytial tissues, Crit. Rev. Biomed. Eng., 21 (1992), 137-199.   Google Scholar

[35]

K. Le Nguyen, Homogenization of heat transfer process in composite materials, J. Elliptic Parabol. Equ., 1 (2015), 175-188.  doi: 10.1007/BF03377374.  Google Scholar

[36]

M. Mabrouk and S. Hassan, Homogenization of a composite medium with a thermal barrier, Math. Meth. Appl. Sci., 27 (2004), 405-425.  doi: 10.1002/mma.460.  Google Scholar

[37]

J. NagumoS. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. Institute of Radio Engineers, 50 (1962), 2061-2070.   Google Scholar

[38]

M. PennacchioG. Savaré and P. C. Franzone, Multiscale modeling for the bioelectric activity of the heart, SIAM J. Math. Anal., 37 (2005), 1333-1370.  doi: 10.1137/040615249.  Google Scholar

[39]

L. Tartar, Problèmes d'homogénéisation dans les équations aux dérivées partielles, in Cours Peccot Collège de France, 1977, partiellement rédigé (ed. H.-c. S. d. F. e. N. dans: F. Murat ed.), Université d’Alger (polycopié), 1977/78. Google Scholar

[40]

C. Timofte, Homogenization results for the calcium dynamics in living cells, Math. Comput. Simul., 133 (2017), 165-174.  doi: 10.1016/j.matcom.2015.06.011.  Google Scholar

[41]

M. Veneroni, Reaction-diffusion systems for the microscopic cellular model of the cardiac electric field, Math. Methods Appl. Sci., 29 (2006) 1631–1661. doi: 10.1002/mma.740.  Google Scholar

[42]

M. Veneroni, Reaction-diffusion systems for the macroscopic bidomain model of the cardiac electric field, Nonlinear Anal. Real World Appl., 10 (2009), 849-868.  doi: 10.1016/j.nonrwa.2007.11.008.  Google Scholar

show all references

References:
[1]

E. AcerbiV. Chiadò PiatG. Dal Maso and D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Anal., 18 (1992), 481-496.  doi: 10.1016/0362-546X(92)90015-7.  Google Scholar

[2]

M. AmarD. Andreucci and D. Bellaveglia, Homogenization of an alternating Robin-Neumann boundary condition via time-periodic unfolding, Nonlinear Anal. Theory Methods Appl., 153 (2017), 56-77.  doi: 10.1016/j.na.2016.05.018.  Google Scholar

[3]

M. AmarD. Andreucci and D. Bellaveglia, The time-periodic unfolding operator and applications to parabolic homogenization, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 28 (2017), 663-700.  doi: 10.4171/RLM/781.  Google Scholar

[4]

M. AmarD. AndreucciP. Bisegna and R. Gianni, Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues, Math. Models Methods Appl. Sci, 14 (2004), 1261-1295.  doi: 10.1142/S0218202504003623.  Google Scholar

[5]

M. AmarD. AndreucciP. Bisegna and R. Gianni, Existence and uniqueness for an elliptic problem with evolution arising in electrodynamics, Nonlinear Anal. Real World Appl., 6 (2005), 367-380.  doi: 10.1016/j.nonrwa.2004.09.002.  Google Scholar

[6]

M. AmarD. AndreucciP. Bisegna and R. Gianni, On a hierarchy of models for electrical conduction in biological tissues, Math. Methods Appl. Sci., 29 (2006), 767-787.  doi: 10.1002/mma.709.  Google Scholar

[7]

M. AmarD. AndreucciP. Bisegna and R. Gianni, A hierarchy of models for the electrical conduction in biological tissues via two-scale convergence: The nonlinear case, Differ. Integral Equ., 26 (2013), 885-912.   Google Scholar

[8]

M. Amar, D. Andreucci, R. Gianni and C. Timofte, Concentration and homogenization in electrical conduction in heterogeneous media involving the Laplace-Beltrami operator, Calc. Var., 59: 99 (2020). doi: 10.1007/s00526-020-01749-x.  Google Scholar

[9]

M. Amar, D. Andreucci and C. Timofte, Well-posedness for a modified bidomain model describing bioelectric activity in damaged heart tissue, preprint, arXiv: 2101.09285. Google Scholar

[10]

M. Amar, I. De Bonis and G. Riey, Homogenization of elliptic problems involving interfaces and singular data, Nonlinear Anal., 189 (2019), 111562. Corrigendum to Homogenization of elliptic problems involving interfaces and singular data. Nonlinear Analysis 203 (2021), 112192. doi: 10.1016/j.na.2020.112192.  Google Scholar

[11]

M. Amar and R. Gianni, Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices, Discrete Contin. Dyn. Systems - B, (4)23 (2018), 1739-1756.  doi: 10.3934/dcdsb.2018078.  Google Scholar

[12]

M. Bendahmane and H. Karlsen, Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue, Netw. Heterog. Media, 1 (2006), 185-218.  doi: 10.3934/nhm.2006.1.185.  Google Scholar

[13]

M. Boulakia, Etude mathématique et numérique de modèles issus du domaine biomédical, Equations aux dérivées partielles, UPMC, 2015. Google Scholar

[14]

M. BoulakiaS. CazeauM. A. FernándezJ. F. Gerbeau and N. Zemzemi, Mathematical modeling of electrocardiograms: a numerical study, Ann. Biomed. Eng., 38 (2010), 1071-1097.   Google Scholar

[15]

M. Boulakia, M. A. Fernández, J. F. Gerbeau and N. Zemzemi, Towards the numerical simulation of electrocardiograms, in Functional Imaging and Modeling of the Heart. FIMH 2007. In Lecture Notes in Computer Science (eds. F. Sachse and G. Seemann), Springer, Berlin, 2007, 240–249. Google Scholar

[16]

Y. BourgaultY. Coudière and C. Pierre, Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology, Nonlinear Anal. Real World Appl., (1) (2009), 458-482.  doi: 10.1016/j.nonrwa.2007.10.007.  Google Scholar

[17]

D. CioranescuA. DamlamianP. DonatoG. Griso and R. Zaki, The periodic unfolding method in domains with holes, SIAM J. Math Anal., 44 (2012), 718-760.  doi: 10.1137/100817942.  Google Scholar

[18]

D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method. Theory and Applications to Partial Differential Problems, Springer, Singapore, 2018. doi: 10.1007/978-981-13-3032-2.  Google Scholar

[19]

D. Cioranescu and J. S. J. Paulin, Homogenization in open sets with holes, J. Math. Anal. Appl., 71 (1979), 590-607.  doi: 10.1016/0022-247X(79)90211-7.  Google Scholar

[20]

A. Collin and S. Imperiale, Mathematical analysis and $2$-scale convergence of an heterogeneous microscopic bidomain model, Math. Models Meth. Appl. Sci., 28 (2018), 979-1035.  doi: 10.1142/S0218202518500264.  Google Scholar

[21]

Y. Coudière, A. Davidovic and C. Poignard, Modified bidomain model with passive periodic heterogeneities, Discrete Contin. Dyn. Systems-S, 13 (2020), 2231-2258. doi: 10.3934/dcdss.2020126.  Google Scholar

[22]

A. Davidovi$\grave{\rm c}$, Multiscale Mathematical Modelling of Structural Heterogeneities in Cardiac Electrophysiology, General Mathematics, Universitè de Bordeaux, 2016. Google Scholar

[23]

P. Donato and K. Le Nguyen, Homogenization for diffusion problems with a nonlinear interfacial resistance, Nonlinear Differ. Equ. Appl., 22 (2015), 1345-1380.  doi: 10.1007/s00030-015-0325-2.  Google Scholar

[24]

A. Gaudiello and M. Lenczner, A two-dimensional electrostatic model of interdigitated comb drive in longitudinal mode, Siam J. Appl. Math., 80 (2020), 792-813.  doi: 10.1137/19M1270306.  Google Scholar

[25]

P. GoelJ. Sneyd and A. Friedman, Homogenization of the cell cytoplasm: The calcium bidomain equations, Multiscale Model. Simul., 5 (2006), 1045-1062.  doi: 10.1137/060660783.  Google Scholar

[26]

I. Graf and M. Peter, Diffusion on surfaces and the boundary periodic unfolding operator with an application to carcinogenesis in human cells, SIAM J. Math. Anal., 46 (2014), 3025-3049.  doi: 10.1137/130921015.  Google Scholar

[27]

I. GrafM. Peter and J. Sneyd, Homogenization of a nonlinear multiscale model of calcium dynamics in biological cells, J. Math. Anal. Appl., 419 (2014), 28-47.  doi: 10.1016/j.jmaa.2014.04.037.  Google Scholar

[28]

E. Grandelius and K. Karlsen, The cardiac bidomain model and homogenization, Netw. Heterog. Media, 14 (2019), 173-204.  doi: 10.3934/nhm.2019009.  Google Scholar

[29]

E. HigginsP. GoelJ. PuglisiD. BersM. Cannell and J. Sneyd, Modelling calcium microdomains using homogenisation, J. Theor. Biol., 247 (2007), 623-644.  doi: 10.1016/j.jtbi.2007.03.019.  Google Scholar

[30]

M. Höpker, Extension operators for Sobolev spaces on periodic domains, their applications, and homogenization of a phase field model for phase transitions in porous media, Ph. D. Thesis, Universit$\ddot{a}$t Bremen, 2016.  Google Scholar

[31]

C. Jerez-HanckesI. Pettersson and V. Rybalko, Derivation of cable equation by multiscale analysis for a model of myelinated axons, Discrete Contin. Dyn. Systems-B, 25 (2020), 815-839.  doi: 10.3934/dcdsb.2019191.  Google Scholar

[32]

N. Kajiwara, On the bidomain equations as parabolic evolution equations, Preprint, available from https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/251618/1/2090-02.pdf. Google Scholar

[33]

J. Keener and J. Sneyd, Mathematical Physiology, Springer, 2004.  Google Scholar

[34]

W. Krassowska and J. Neu, Homogenization of syncytial tissues, Crit. Rev. Biomed. Eng., 21 (1992), 137-199.   Google Scholar

[35]

K. Le Nguyen, Homogenization of heat transfer process in composite materials, J. Elliptic Parabol. Equ., 1 (2015), 175-188.  doi: 10.1007/BF03377374.  Google Scholar

[36]

M. Mabrouk and S. Hassan, Homogenization of a composite medium with a thermal barrier, Math. Meth. Appl. Sci., 27 (2004), 405-425.  doi: 10.1002/mma.460.  Google Scholar

[37]

J. NagumoS. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. Institute of Radio Engineers, 50 (1962), 2061-2070.   Google Scholar

[38]

M. PennacchioG. Savaré and P. C. Franzone, Multiscale modeling for the bioelectric activity of the heart, SIAM J. Math. Anal., 37 (2005), 1333-1370.  doi: 10.1137/040615249.  Google Scholar

[39]

L. Tartar, Problèmes d'homogénéisation dans les équations aux dérivées partielles, in Cours Peccot Collège de France, 1977, partiellement rédigé (ed. H.-c. S. d. F. e. N. dans: F. Murat ed.), Université d’Alger (polycopié), 1977/78. Google Scholar

[40]

C. Timofte, Homogenization results for the calcium dynamics in living cells, Math. Comput. Simul., 133 (2017), 165-174.  doi: 10.1016/j.matcom.2015.06.011.  Google Scholar

[41]

M. Veneroni, Reaction-diffusion systems for the microscopic cellular model of the cardiac electric field, Math. Methods Appl. Sci., 29 (2006) 1631–1661. doi: 10.1002/mma.740.  Google Scholar

[42]

M. Veneroni, Reaction-diffusion systems for the macroscopic bidomain model of the cardiac electric field, Nonlinear Anal. Real World Appl., 10 (2009), 849-868.  doi: 10.1016/j.nonrwa.2007.11.008.  Google Scholar

Figure 1.  ${Left}$: the periodic cell $ Y $. $ E^{ \rm{D}} $ is the shaded region and $ E^{ \rm{B}} $ is the white region.${Right}$: the region $ \varOmega $
Figure 2.  The periodic cell $ Y $. $ E^{ \rm{D}} $ is the shaded region and $ E^{ \rm{B}} $ is the white region
[1]

Zhanying Yang. Homogenization and correctors for the hyperbolic problems with imperfect interfaces via the periodic unfolding method. Communications on Pure & Applied Analysis, 2014, 13 (1) : 249-272. doi: 10.3934/cpaa.2014.13.249

[2]

Erik Grandelius, Kenneth H. Karlsen. The cardiac bidomain model and homogenization. Networks & Heterogeneous Media, 2019, 14 (1) : 173-204. doi: 10.3934/nhm.2019009

[3]

Seiji Ukai. Time-periodic solutions of the Boltzmann equation. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 579-596. doi: 10.3934/dcds.2006.14.579

[4]

Jean-Jérôme Casanova. Existence of time-periodic strong solutions to a fluid–structure system. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3291-3313. doi: 10.3934/dcds.2019136

[5]

Taige Wang, Bing-Yu Zhang. Forced oscillation of viscous Burgers' equation with a time-periodic force. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1205-1221. doi: 10.3934/dcdsb.2020160

[6]

Martin Heida, Alexander Mielke. Averaging of time-periodic dissipation potentials in rate-independent processes. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1303-1327. doi: 10.3934/dcdss.2017070

[7]

Peter Giesl, Holger Wendland. Approximating the basin of attraction of time-periodic ODEs by meshless collocation. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1249-1274. doi: 10.3934/dcds.2009.25.1249

[8]

Yi Wang, Dun Zhou. Transversality for time-periodic competitive-cooperative tridiagonal systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1821-1830. doi: 10.3934/dcdsb.2015.20.1821

[9]

Xinjian Wang, Guo Lin. Asymptotic spreading for a time-periodic predator-prey system. Communications on Pure & Applied Analysis, 2019, 18 (6) : 2983-2999. doi: 10.3934/cpaa.2019133

[10]

Fioralba Cakoni, Houssem Haddar, Isaac Harris. Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem. Inverse Problems & Imaging, 2015, 9 (4) : 1025-1049. doi: 10.3934/ipi.2015.9.1025

[11]

Pavao Mardešić, David Marín, Jordi Villadelprat. Unfolding of resonant saddles and the Dulac time. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1221-1244. doi: 10.3934/dcds.2008.21.1221

[12]

Morteza Fotouhi, Mohsen Yousefnezhad. Homogenization of a locally periodic time-dependent domain. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1669-1695. doi: 10.3934/cpaa.2020061

[13]

Xiongxiong Bao, Wan-Tong Li, Zhi-Cheng Wang. Uniqueness and stability of time-periodic pyramidal fronts for a periodic competition-diffusion system. Communications on Pure & Applied Analysis, 2020, 19 (1) : 253-277. doi: 10.3934/cpaa.2020014

[14]

Yves Coudière, Anđela Davidović, Clair Poignard. Modified bidomain model with passive periodic heterogeneities. Discrete & Continuous Dynamical Systems - S, 2020, 13 (8) : 2231-2258. doi: 10.3934/dcdss.2020126

[15]

Wei-Jie Sheng, Wan-Tong Li. Multidimensional stability of time-periodic planar traveling fronts in bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2681-2704. doi: 10.3934/dcds.2017115

[16]

Petr Kučera. The time-periodic solutions of the Navier-Stokes equations with mixed boundary conditions. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 325-337. doi: 10.3934/dcdss.2010.3.325

[17]

Peter Giesl, Holger Wendland. Approximating the basin of attraction of time-periodic ODEs by meshless collocation of a Cauchy problem. Conference Publications, 2009, 2009 (Special) : 259-268. doi: 10.3934/proc.2009.2009.259

[18]

Ke Wang, Qi Wang, Feng Yu. Stationary and time-periodic patterns of two-predator and one-prey systems with prey-taxis. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 505-543. doi: 10.3934/dcds.2017021

[19]

Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3063-3092. doi: 10.3934/dcds.2020398

[20]

Meng Zhao, Wan-Tong Li, Jia-Feng Cao. A prey-predator model with a free boundary and sign-changing coefficient in time-periodic environment. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3295-3316. doi: 10.3934/dcdsb.2017138

2019 Impact Factor: 1.105

Article outline

Figures and Tables

[Back to Top]