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On the Bidomain equations driven by stochastic forces

  • * Corresponding author: Oleksandr Misiats

    * Corresponding author: Oleksandr Misiats 
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  • 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.

    Mathematics Subject Classification: Primary: 35R60, 60H15, 92C35.


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  • [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. BourgaultY. 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-FranzoneL. 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 PratoS. 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. ZabczykStochastic Equations in Infinite Dimensions, Cambridge Univerity Press, 1992.  doi: 10.1017/CBO9780511666223.
    [9] G. Da Prato and  J. ZabczykErgodicity 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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