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November  2020, 40(11): 6159-6177. doi: 10.3934/dcds.2020274

On the Bidomain equations driven by stochastic forces

Matthias Hieber 1, , Oleksandr Misiats 2,, and  Oleksandr Stanzhytskyi 3,

1. 

Technische Universität Darmstadt, Fachbereich Mathematik, Schlossgarten-Strasse 7, D-64289 Darmstadt, Germany
2. 

Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, 1015 Floyd ave., Richmond, VA, 23227, USA
3. 

Department of Mathematics, Taras Shevchenko National University of Kyiv, 4E Glushkov ave., Kyiv, 03127, Ukraine

* Corresponding author: Oleksandr Misiats

Received  September 2019 Revised  June 2020 Published  July 2020

The bidomain equations driven by stochastic forces and subject to nonlinearities of FitzHugh-Nagumo or Allen-Cahn type are considered for the first time. It is shown that this set of equations admits a global weak solution as well as a stationary solution, which generates a uniquely determined invariant measure.

Keywords: Bidomain equations, white noise, invariant measure.
Mathematics Subject Classification: Primary: 35R60, 60H15, 92C35.
Citation: Matthias Hieber, Oleksandr Misiats, Oleksandr Stanzhytskyi. On the Bidomain equations driven by stochastic forces. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6159-6177. doi: 10.3934/dcds.2020274
References:
[1]

R. R. Aliev and A. V. Panfilov, A simple two-variable model for cardiac excitation, Fractals, 7 (1996), 293-301.  doi: 10.1016/0960-0779(95)00089-5.

[2]

Y. Bourgault, Y. Coudière and C. Pierre, Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology, Nonlinear Anal. Real World Appl., 10 (2009), 458-482.  doi: 10.1016/j.nonrwa.2007.10.007.

[3]

P.-L. Chow, Stochastic Partial Differential Equations, Chapman & Hall/CRC, Boca Raton, FL, 2007.

[4]

P. Colli-Franzone, L. Guerri and S. Tentoni, Mathematical modeling of the excitation process in myocardium tissues: Influence of fiber rotation on wavefront progagation and potential field, Math. Biosci., 110 (1990), 155-235. 

[5]

P. Colli Franzone, L. F. Pavarino and S. Scacchi, Mathematical Cardiac Electrophysiology, Springer, 2014. doi: 10.1007/978-3-319-04801-7.

[6]

P. Colli Franzone and G. Savaré, Degenerate evolution systems modeling the cardiac electric field at micro- and macroscopic level, Evolution Equations, Semigroups and Functional Analysis. Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 50 (2002), 49–78.

[7]

G. Da Prato, S. Kwapień and J. Zabczyk, Regularity of solutions of linear stochastic equations in Hilbert spaces, Stochastics, 23 (1987), 1-23.  doi: 10.1080/17442508708833480.

[8] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univerity Press, 1992.  doi: 10.1017/CBO9780511666223.
[9] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge Univerity Press, 1996.  doi: 10.1017/CBO9780511662829.
[10]

Y. Giga and N. Kajiwara, On a resolvent estimate for bidomain operators and its applications, J. Math. Anal. Appl., 459 (2018), 528-555.  doi: 10.1016/j.jmaa.2017.10.023.

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 1981.

[12]

M. Hieber and J. Prüss, $L_q$-Theory for the bidomain operator, Submitted.

[13]

M. Hieber and J. Prüss, On the bidomain problem with FitzHugh-Nagumo transport, Arch. Math., 111 (2018), 313–327. doi: 10.1007/s00013-018-1188-7.

[14]

J. Keener and J. Sneyd, Mathematical Physiology, Interdisciplinary Applied Mathematics Springer, New York, 1998.

[15]

O. Misiats, O. Stanzhytskyi and N. K. Yip, Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains, Journal of Theoretical Probability, 29 (2016), 996–1026. doi: 10.1007/s10959-015-0606-z.

[16]

O. Misiats, O. Stanzhytskyi and N. K. Yip, Asymptotic behavior and homogenization of invariant measures, Stoch. Dyn., 19 (2019), 1950015, 27 pp. doi: 10.1142/S0219493719500151.

[17]

O. Misiats, O. Stanzhytskyi and N. K. Yip, Invariant measures for reaction-diffusion equations with weakly dissipative nonlinearities, Stochastics: An International Journal of Probability and Stochastic Processes, 2019. doi: 10.1080/17442508.2019.1691212.

[18]

Y. Mori and H. Matano, Stability of front solutions of the bidomain equation, Comm. Pure Appl. Math., 69 (2016), 2364-2426.  doi: 10.1002/cpa.21634.

[19]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics, 2007.

[20]

J. Rogers and A. McCulloch, A collacation Galerkin finite element model of cardiac action potential propagation, IEEE Trans. Biomed. Eng., 41 (1994), 743–757.

[21]

H. Tanabe, Equations of Evolution, Pitman, 1979.

[22]

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.

show all references

References:
[1]

R. R. Aliev and A. V. Panfilov, A simple two-variable model for cardiac excitation, Fractals, 7 (1996), 293-301.  doi: 10.1016/0960-0779(95)00089-5.

[2]

Y. Bourgault, Y. Coudière and C. Pierre, Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology, Nonlinear Anal. Real World Appl., 10 (2009), 458-482.  doi: 10.1016/j.nonrwa.2007.10.007.

[3]

P.-L. Chow, Stochastic Partial Differential Equations, Chapman & Hall/CRC, Boca Raton, FL, 2007.

[4]

P. Colli-Franzone, L. Guerri and S. Tentoni, Mathematical modeling of the excitation process in myocardium tissues: Influence of fiber rotation on wavefront progagation and potential field, Math. Biosci., 110 (1990), 155-235. 

[5]

P. Colli Franzone, L. F. Pavarino and S. Scacchi, Mathematical Cardiac Electrophysiology, Springer, 2014. doi: 10.1007/978-3-319-04801-7.

[6]

P. Colli Franzone and G. Savaré, Degenerate evolution systems modeling the cardiac electric field at micro- and macroscopic level, Evolution Equations, Semigroups and Functional Analysis. Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 50 (2002), 49–78.

[7]

G. Da Prato, S. Kwapień and J. Zabczyk, Regularity of solutions of linear stochastic equations in Hilbert spaces, Stochastics, 23 (1987), 1-23.  doi: 10.1080/17442508708833480.

[8] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univerity Press, 1992.  doi: 10.1017/CBO9780511666223.
[9] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge Univerity Press, 1996.  doi: 10.1017/CBO9780511662829.
[10]

Y. Giga and N. Kajiwara, On a resolvent estimate for bidomain operators and its applications, J. Math. Anal. Appl., 459 (2018), 528-555.  doi: 10.1016/j.jmaa.2017.10.023.

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 1981.

[12]

M. Hieber and J. Prüss, $L_q$-Theory for the bidomain operator, Submitted.

[13]

M. Hieber and J. Prüss, On the bidomain problem with FitzHugh-Nagumo transport, Arch. Math., 111 (2018), 313–327. doi: 10.1007/s00013-018-1188-7.

[14]

J. Keener and J. Sneyd, Mathematical Physiology, Interdisciplinary Applied Mathematics Springer, New York, 1998.

[15]

O. Misiats, O. Stanzhytskyi and N. K. Yip, Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains, Journal of Theoretical Probability, 29 (2016), 996–1026. doi: 10.1007/s10959-015-0606-z.

[16]

O. Misiats, O. Stanzhytskyi and N. K. Yip, Asymptotic behavior and homogenization of invariant measures, Stoch. Dyn., 19 (2019), 1950015, 27 pp. doi: 10.1142/S0219493719500151.

[17]

O. Misiats, O. Stanzhytskyi and N. K. Yip, Invariant measures for reaction-diffusion equations with weakly dissipative nonlinearities, Stochastics: An International Journal of Probability and Stochastic Processes, 2019. doi: 10.1080/17442508.2019.1691212.

[18]

Y. Mori and H. Matano, Stability of front solutions of the bidomain equation, Comm. Pure Appl. Math., 69 (2016), 2364-2426.  doi: 10.1002/cpa.21634.

[19]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics, 2007.

[20]

J. Rogers and A. McCulloch, A collacation Galerkin finite element model of cardiac action potential propagation, IEEE Trans. Biomed. Eng., 41 (1994), 743–757.

[21]

H. Tanabe, Equations of Evolution, Pitman, 1979.

[22]

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.

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