# American Institute of Mathematical Sciences

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

## On the Bidomain equations driven by stochastic forces

 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.

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.

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##### 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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