
Previous Article
Numerical simulation of resonant tunneling of fast solitons for the nonlinear Schrödinger equation
 DCDS Home
 This Issue

Next Article
Delone measures of finite local complexity and applications to spectral theory of onedimensional continuum models of quasicrystals
Global existence and uniqueness of a threedimensional model of cellular electrophysiology
1.  Graduate School of Mathematical Sciences, University of Tokyo, Komaba Tokyo, 1538914 
2.  School of Mathematics, University of Minnesota, 206 Church St. SE, Minneapolis MN, 55414, United States 
References:
[1] 
R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," Academic Press, 2003. 
[2] 
M. Amar, D. Andreucci, P. Bisegna and R. Gianni, Existence and uniqueness for an elliptic problem with evolution arising in electrodynamics, Nonlinear Analysis: Real World Applications, 6 (2005), 367380. doi: 10.1016/j.nonrwa.2004.09.002. 
[3] 
W. Arendt, Oneparameter semigroups of positive operators, Lecture Notes in Mathematics, vol. 1184, ch. BII, "Characterization of Positive Semigroups on $C_0(X)$," SpringerVerlag, 1980. 
[4] 
V. Barcilon, J. D. Cole and R. S. Eisenberg, A singular perturbation analysis of induced electric fields in nerve cells, SIAM Journal on Applied Mathematics, 21 (1971), 339354. doi: 10.1137/0121036. 
[5] 
E. B. Davies, "Heat Kernels and Spectral Theory," Cambridge University Press, 1990. 
[6] 
R. S. Eisenberg and E. A. Johnson, Threedimensional electrical field problems in physiology, Prog. Biophys. Mol. Biol, 20 (1970), 165. doi: 10.1016/00796107(70)900131. 
[7] 
J. Escher, Nonlinear elliptic systems with dynamic boundary conditions, Mathematische Zeitschrift, 210 (1992), 413439. doi: 10.1007/BF02571805. 
[8] 
G. B. Folland, "Introduction to Partial Differential Equations," Princeton University Press, 1995. 
[9] 
P. C. Franzone and G. Savare, Degenerate evolution systems modeling the cardiac electric field at micro and macroscopic level, Evolution Equations, Semigroups and Functional Analysis: In memory of Brunello Terreni, 50 (2002), 4978. 
[10] 
C. Gold, D. A. Henze, C. Koch and G. Buzsaki, On the origin of the extracellular action potential waveform: A modeling study, Journal of Neurophysiology, 95 (2006), 31133128. doi: 10.1152/jn.00979.2005. 
[11] 
J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Amer. Mathematical Society, 1988. 
[12] 
T. Hintermann, Evolution equations with dynamic boundary conditions, Proc. Royal Soc. Edinburgh A, 113 (1989), 4360. 
[13] 
G. R. Holt and C. Koch, Electrical interactions via the extracellular potential near cell bodies, Journal of Computational Neuroscience, 6 (1999), 169184. doi: 10.1023/A:1008832702585. 
[14] 
J. P. Keener and J. Sneyd, "Mathematical Physiology," SpringerVerlag, New York, 1998. 
[15] 
C. Koch, "Biophysics of Computation," Oxford University Press, New York, 1999. 
[16] 
J. Lee and G. Uhlmann, Determining anisotropic realanalytic conductivities by boundary measurements, Comm. Pure Appl. Math, 42 (1989), 10971112. doi: 10.1002/cpa.3160420804. 
[17] 
M. Léonetti, E. DuboisViolette and F. Homblé, Pattern formation of stationaly transcellular ionic currents in Fucus, Proc. Natl. Acad. Sci. USA, 101 (2004), 1024310248. doi: 10.1073/pnas.0402335101. 
[18] 
J. L. Lions and M. E., "NonHomogeneous Boundary Value Problems and Applications," SpringerVerlag New York, 1972. 
[19] 
A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Birkhäuser, 1995. 
[20] 
J. MalletParet, Negatively invariant sets of compact maps and an extension of a theorem of Cartwright, Journal of Differential Equations, 22 (1976), 331348. doi: 10.1016/00220396(76)900322. 
[21] 
R. Mané, "On the Dimension of the Compact Invariant Sets of Certain NonLinear Maps," Dynamical Systems and Turbulence, Warwick 1980 (1981), 230242. doi: 10.1007/BFb0091916. 
[22] 
M. Marion, Finitedimensional attractors associated with partly dissipative reactiondiffusion systems, SIAM Journal on Mathematical Analysis, 20 (1989), 816. doi: 10.1137/0520057. 
[23] 
Y. Mori, G. I. Fishman and C. S. Peskin, Ephaptic conduction in a cardiac strand model with 3d electrodiffusion, Proceedings of the National Academy of Sciences, 105 (2008), 64636468. doi: 10.1073/pnas.0801089105. 
[24] 
Y. Mori, J. W. Jerome and C. S. Peskin, "A ThreeDimensional Model of Cellular Electrical Activity," Bulletin of the Institute of Mathematics, Academia Sinica, Taiwan, 2007. 
[25] 
J. C. Neu and W. Krassowska, Homogenization of syncytial tissues, Critical Reviews in Biomedical Engineering, 21 (1993), 137199. 
[26] 
E. M. Ouhabaz, "Analysis of Heat Equations on Domains," London Mathematical Society Monographs, vol. 31, Princeton University Press, 2005. 
[27] 
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer, 1983. 
[28] 
M. Pennacchio, G. Savare and P. C. Franzone, Multiscale modeling for the bioelectric activity of the heart, SIAM Journal on Mathematical Analysis, 37 (2006), 1333. doi: 10.1137/040615249. 
[29] 
W. Rall, Distribution of potential in cylindrical coordinates and time constants for a membrane cylinder, Biophys. J., 9 (1969), 15091541. doi: 10.1016/S00063495(69)864684. 
[30] 
J. Rauch and J. Smoller, Qualitative theory of the fitzhughnagumo equations, Advances in Mathematics, 27 (1978), 1244. doi: 10.1016/00018708(78)900750. 
[31]  
[32] 
T. Runst and W. Sickel, "Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations," Walter de Gruyter, 1996. 
[33] 
G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Springer Verlag, 2002. 
[34] 
J. Smoller, "Shock Waves and ReactionDiffusion Equations," Grundlehren der mathematischen Wissenschaften, vol. 258, SpringerVerlag, 1994. 
[35] 
M. E. Taylor, "Partial Differential Equations, vol. I, II, III," SpringerVerlag, 1996. 
[36] 
M. Veneroni, Reaction diffusion systems for the microscopic cellular model of the cardiac electric field, Mathematical Methods in the Applied Sciences, 29 (2006), 16311661. doi: 10.1002/mma.740. 
show all references
References:
[1] 
R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," Academic Press, 2003. 
[2] 
M. Amar, D. Andreucci, P. Bisegna and R. Gianni, Existence and uniqueness for an elliptic problem with evolution arising in electrodynamics, Nonlinear Analysis: Real World Applications, 6 (2005), 367380. doi: 10.1016/j.nonrwa.2004.09.002. 
[3] 
W. Arendt, Oneparameter semigroups of positive operators, Lecture Notes in Mathematics, vol. 1184, ch. BII, "Characterization of Positive Semigroups on $C_0(X)$," SpringerVerlag, 1980. 
[4] 
V. Barcilon, J. D. Cole and R. S. Eisenberg, A singular perturbation analysis of induced electric fields in nerve cells, SIAM Journal on Applied Mathematics, 21 (1971), 339354. doi: 10.1137/0121036. 
[5] 
E. B. Davies, "Heat Kernels and Spectral Theory," Cambridge University Press, 1990. 
[6] 
R. S. Eisenberg and E. A. Johnson, Threedimensional electrical field problems in physiology, Prog. Biophys. Mol. Biol, 20 (1970), 165. doi: 10.1016/00796107(70)900131. 
[7] 
J. Escher, Nonlinear elliptic systems with dynamic boundary conditions, Mathematische Zeitschrift, 210 (1992), 413439. doi: 10.1007/BF02571805. 
[8] 
G. B. Folland, "Introduction to Partial Differential Equations," Princeton University Press, 1995. 
[9] 
P. C. Franzone and G. Savare, Degenerate evolution systems modeling the cardiac electric field at micro and macroscopic level, Evolution Equations, Semigroups and Functional Analysis: In memory of Brunello Terreni, 50 (2002), 4978. 
[10] 
C. Gold, D. A. Henze, C. Koch and G. Buzsaki, On the origin of the extracellular action potential waveform: A modeling study, Journal of Neurophysiology, 95 (2006), 31133128. doi: 10.1152/jn.00979.2005. 
[11] 
J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Amer. Mathematical Society, 1988. 
[12] 
T. Hintermann, Evolution equations with dynamic boundary conditions, Proc. Royal Soc. Edinburgh A, 113 (1989), 4360. 
[13] 
G. R. Holt and C. Koch, Electrical interactions via the extracellular potential near cell bodies, Journal of Computational Neuroscience, 6 (1999), 169184. doi: 10.1023/A:1008832702585. 
[14] 
J. P. Keener and J. Sneyd, "Mathematical Physiology," SpringerVerlag, New York, 1998. 
[15] 
C. Koch, "Biophysics of Computation," Oxford University Press, New York, 1999. 
[16] 
J. Lee and G. Uhlmann, Determining anisotropic realanalytic conductivities by boundary measurements, Comm. Pure Appl. Math, 42 (1989), 10971112. doi: 10.1002/cpa.3160420804. 
[17] 
M. Léonetti, E. DuboisViolette and F. Homblé, Pattern formation of stationaly transcellular ionic currents in Fucus, Proc. Natl. Acad. Sci. USA, 101 (2004), 1024310248. doi: 10.1073/pnas.0402335101. 
[18] 
J. L. Lions and M. E., "NonHomogeneous Boundary Value Problems and Applications," SpringerVerlag New York, 1972. 
[19] 
A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Birkhäuser, 1995. 
[20] 
J. MalletParet, Negatively invariant sets of compact maps and an extension of a theorem of Cartwright, Journal of Differential Equations, 22 (1976), 331348. doi: 10.1016/00220396(76)900322. 
[21] 
R. Mané, "On the Dimension of the Compact Invariant Sets of Certain NonLinear Maps," Dynamical Systems and Turbulence, Warwick 1980 (1981), 230242. doi: 10.1007/BFb0091916. 
[22] 
M. Marion, Finitedimensional attractors associated with partly dissipative reactiondiffusion systems, SIAM Journal on Mathematical Analysis, 20 (1989), 816. doi: 10.1137/0520057. 
[23] 
Y. Mori, G. I. Fishman and C. S. Peskin, Ephaptic conduction in a cardiac strand model with 3d electrodiffusion, Proceedings of the National Academy of Sciences, 105 (2008), 64636468. doi: 10.1073/pnas.0801089105. 
[24] 
Y. Mori, J. W. Jerome and C. S. Peskin, "A ThreeDimensional Model of Cellular Electrical Activity," Bulletin of the Institute of Mathematics, Academia Sinica, Taiwan, 2007. 
[25] 
J. C. Neu and W. Krassowska, Homogenization of syncytial tissues, Critical Reviews in Biomedical Engineering, 21 (1993), 137199. 
[26] 
E. M. Ouhabaz, "Analysis of Heat Equations on Domains," London Mathematical Society Monographs, vol. 31, Princeton University Press, 2005. 
[27] 
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer, 1983. 
[28] 
M. Pennacchio, G. Savare and P. C. Franzone, Multiscale modeling for the bioelectric activity of the heart, SIAM Journal on Mathematical Analysis, 37 (2006), 1333. doi: 10.1137/040615249. 
[29] 
W. Rall, Distribution of potential in cylindrical coordinates and time constants for a membrane cylinder, Biophys. J., 9 (1969), 15091541. doi: 10.1016/S00063495(69)864684. 
[30] 
J. Rauch and J. Smoller, Qualitative theory of the fitzhughnagumo equations, Advances in Mathematics, 27 (1978), 1244. doi: 10.1016/00018708(78)900750. 
[31]  
[32] 
T. Runst and W. Sickel, "Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations," Walter de Gruyter, 1996. 
[33] 
G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Springer Verlag, 2002. 
[34] 
J. Smoller, "Shock Waves and ReactionDiffusion Equations," Grundlehren der mathematischen Wissenschaften, vol. 258, SpringerVerlag, 1994. 
[35] 
M. E. Taylor, "Partial Differential Equations, vol. I, II, III," SpringerVerlag, 1996. 
[36] 
M. Veneroni, Reaction diffusion systems for the microscopic cellular model of the cardiac electric field, Mathematical Methods in the Applied Sciences, 29 (2006), 16311661. doi: 10.1002/mma.740. 
[1] 
Ning Ju. The global attractor for the solutions to the 3D viscous primitive equations. Discrete and Continuous Dynamical Systems, 2007, 17 (1) : 159179. doi: 10.3934/dcds.2007.17.159 
[2] 
Ning Ju. The finite dimensional global attractor for the 3D viscous Primitive Equations. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 70017020. doi: 10.3934/dcds.2016104 
[3] 
T. Tachim Medjo. A nonautonomous 3D Lagrangian averaged NavierStokes$\alpha$ model with oscillating external force and its global attractor. Communications on Pure and Applied Analysis, 2011, 10 (2) : 415433. doi: 10.3934/cpaa.2011.10.415 
[4] 
Boyan Jonov, Thomas C. Sideris. Global and almost global existence of small solutions to a dissipative wave equation in 3D with nearly null nonlinear terms. Communications on Pure and Applied Analysis, 2015, 14 (4) : 14071442. doi: 10.3934/cpaa.2015.14.1407 
[5] 
Mohamad Darwich. Local and global wellposedness in the energy space for the dissipative ZakharovKuznetsov equation in 3D. Discrete and Continuous Dynamical Systems  B, 2020, 25 (9) : 37153724. doi: 10.3934/dcdsb.2020087 
[6] 
Yong Yang, Bingsheng Zhang. On the Kolmogorov entropy of the weak global attractor of 3D NavierStokes equations:Ⅰ. Discrete and Continuous Dynamical Systems  B, 2017, 22 (6) : 23392350. doi: 10.3934/dcdsb.2017101 
[7] 
Boling Guo, Guoli Zhou. Finite dimensionality of global attractor for the solutions to 3D viscous primitive equations of largescale moist atmosphere. Discrete and Continuous Dynamical Systems  B, 2018, 23 (10) : 43054327. doi: 10.3934/dcdsb.2018160 
[8] 
T. Tachim Medjo. Nonautonomous 3D primitive equations with oscillating external force and its global attractor. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 265291. doi: 10.3934/dcds.2012.32.265 
[9] 
Ciprian G. Gal, T. Tachim Medjo. Approximation of the trajectory attractor for a 3D model of incompressible twophaseflows. Communications on Pure and Applied Analysis, 2014, 13 (6) : 22292252. doi: 10.3934/cpaa.2014.13.2229 
[10] 
Vladimir V. Chepyzhov, E. S. Titi, Mark I. Vishik. On the convergence of solutions of the Leray$\alpha $ model to the trajectory attractor of the 3D NavierStokes system. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 481500. doi: 10.3934/dcds.2007.17.481 
[11] 
Gabriel Deugoue. Approximation of the trajectory attractor of the 3D MHD System. Communications on Pure and Applied Analysis, 2013, 12 (5) : 21192144. doi: 10.3934/cpaa.2013.12.2119 
[12] 
M. Bulíček, F. Ettwein, P. Kaplický, Dalibor Pražák. The dimension of the attractor for the 3D flow of a nonNewtonian fluid. Communications on Pure and Applied Analysis, 2009, 8 (5) : 15031520. doi: 10.3934/cpaa.2009.8.1503 
[13] 
Gerhard Kirsten. Multilinear PODDEIM model reduction for 2D and 3D semilinear systems of differential equations. Journal of Computational Dynamics, 2022, 9 (2) : 159183. doi: 10.3934/jcd.2021025 
[14] 
Xiaojing Xu, Zhuan Ye. Note on global regularity of 3D generalized magnetohydrodynamic$\alpha$ model with zero diffusivity. Communications on Pure and Applied Analysis, 2015, 14 (2) : 585595. doi: 10.3934/cpaa.2015.14.585 
[15] 
Edriss S. Titi, Saber Trabelsi. Global wellposedness of a 3D MHD model in porous media. Journal of Geometric Mechanics, 2019, 11 (4) : 621637. doi: 10.3934/jgm.2019031 
[16] 
Jingrui Su. Global existence and low Mach number limit to a 3D compressible micropolar fluids model in a bounded domain. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 34233434. doi: 10.3934/dcds.2017145 
[17] 
Makram Hamouda, ChangYeol Jung, Roger Temam. Asymptotic analysis for the 3D primitive equations in a channel. Discrete and Continuous Dynamical Systems  S, 2013, 6 (2) : 401422. doi: 10.3934/dcdss.2013.6.401 
[18] 
Junxiong Jia, Jigen Peng, Kexue Li. On the decay and stability of global solutions to the 3D inhomogeneous MHD system. Communications on Pure and Applied Analysis, 2017, 16 (3) : 745780. doi: 10.3934/cpaa.2017036 
[19] 
Xiaojie Yang, Hui Liu, Chengfeng Sun. Global attractors of the 3D micropolar equations with damping term. Mathematical Foundations of Computing, 2021, 4 (2) : 117130. doi: 10.3934/mfc.2021007 
[20] 
Anthony Suen. Global regularity for the 3D compressible magnetohydrodynamics with general pressure. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 29272943. doi: 10.3934/dcds.2022004 
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]