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Solutions to muscle fiber equations and their long time behaviour. (English) Zbl 1105.35306
Summary: We consider the nonlinear initial-boundary value problem governing the dynamical displacements of a one-dimensional solid body with specific stress–strain law. This constitutive law results from the modelization of the mechanisms that rules the electrically activated mechanical behaviour of cardiac muscle fibers at the microscopic level. We prove global existence and uniqueness of solutions and we study their asymptotic behaviour in time. In particular we show that under vanishing external forcing solutions asymptotically converge to an equilibrium.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
74L15 Biomechanical solid mechanics
92C10 Biomechanics
35B35 Stability in context of PDEs
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[1] Andrews, G., On the existence of solutions to the equation \(u_{\mathit{tt}} = u_{\mathit{xxt}} + \sigma(u_x)_x\), J. diff. equations, 35, 200-231, (1980) · Zbl 0415.35018
[2] Bestel, J.; Clément, F.; Sorine, M., A biomechanical model of muscle contraction, () · Zbl 1041.68560
[3] Chapelle, D.; Clément, F.; Génot, F.; Le Tallec, P.; Sorine, M.; Urquiza, J.M., A physiologically-based model for the active cardiac muscle, (), 128-133 · Zbl 1052.68824
[4] Colli, P., On a nonlinear and nonlocal evolution equation related to muscle contraction, Nonl. anal. theory, methods and appl., 13, 1149-1162, (1989) · Zbl 0685.45010
[5] Comincioli, V.; Torelli, A., A mathematical model of contracting muscle with viscoelastic elements, SIAM J. math. anal., 19, 593-612, (1988) · Zbl 0721.92007
[6] Fung, Y.C., Comparison of different models of the heart muscle, J. biomech., 4, 289-295, (1971)
[7] Hill, A.V., The heat of shortening and the dynamic constants in muscle, Proc. roy. soc. London (B), 126, 136-195, (1938)
[8] Hill, A.V., Mechanics of the contractile element of muscle, Nature, 166, 415-419, (1950)
[9] Hill, A.V., The series elastic component of muscle, Proc. roy. soc. London (B), 137, 272-280, (1950)
[10] Hill, T.L., Theoretical formalism for the sliding filament model of contraction of striated muscle. part I, Prog. biophys. mol. biol., 28, 267-340, (1974)
[11] Hill, T.L., Theoretical formalism for the sliding filament model of contraction of striated muscle. part II, Prog. biophys. mol. biol., 29, 105-159, (1975)
[12] P.J. Hunter, M.P. Nash, G.B. Sands, Computational electromechanics of the heart, in: A.V. Panfilov, A.V. Holden (Eds.), Computational biology of the heart, Wiley, 1997, pp. 345-407. · Zbl 0905.92005
[13] A.F. Huxley, Muscle structure and theories of contraction, in: Progress in biophysics and biological chemistry, vol. 7, Pergamon Press, 1957, pp. 255-318.
[14] Jülicher, F.; Ajdari, A.; Prost, J., Modeling molecular motors, Rev. mod. phys., 69, 4, (1997)
[15] Keener, J.; Sneyd, J., Mathematical physiology, (1998), Springer Berlin · Zbl 0913.92009
[16] P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakuto Int. Series Math. Sci. & Appl., vol. 6, Gakkōtosho, Tokyo, 1996.
[17] Mirsky, I.; Parmley, W.W., Assessment of passive elastic stiffness for isolated heart muscle and the intact heart, Circul. res., 33, 233-243, (1973)
[18] Visintin, A., Differential models of hysteresis, (1994), Springer Berlin · Zbl 0820.35004
[19] Wu, J.Z.; Herzog, W., Modelling concentric contraction of muscle using an improved cross-bridge model, J. biomech., 32, 837-848, (1999)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.