Solutions to muscle fiber equations and their long time behaviour

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.