A nonlinear nonlocal transport system related to the cross-bridge mechanism of muscle contraction (Q1586539)

In this very interesting paper the authors study a nonlocal transport equation of the form \[ u_t+z'(t)u_x=F(x,t,z(t),u)-G(x,t,z(t),u)u, \quad (x,t)\in \mathbb{R}\times(0,\tau) \] \[ z(t)=L\left(\int_{\mathbb{R}}w(x) u(x,t)dx\right), \quad t\in [0,\tau] \] \[ u(x,0)=u_0(x), \quad 0\leq u_0(x)\leq 1 \quad \text{ for a.e.} \quad x\in\mathbb{R}. \] which represents the mathematical model of the cross-bridge mechanism of muscle contraction. Rewriting the above system as an abstract system of integral equations and using suitable conditions upon \(F\) and \(G\) and nonlinear evolution operators techniques, they prove an existence and uniqueness result as well as a continuous dependence theorem upon the data. Under some natural extra conditions on \(F\) and \(G\) they show that \((u(t),z(t))\) tends to \((0,0)\) in \(L^1(\mathbb{R})\times \mathbb{R}\) as \(t\to \infty\).