On the asymptotic spatial behaviour of the solutions of the nerve system

In this paper we investigate the asymptotic spatial behavior of the solutions for several models for the nerve fibers. First, our analysis deals with the coupling of two parabolic equations. We prove that, under suitable assumptions on the coefficients and the nonlinear function, the decay is similar to the one corresponding to the heat equation. A limit case of this system corresponds to the coupling of a parabolic equation with an ordinary differential equation. In this situation, we see that for suitable boundary conditions the solution ceases to exist for a finite value of the spatial variable. Next two sections correspond to the coupling of a hyperbolic/parabolic and hyperbolic/ordinary differential problems. For the first one we obtain that the decay is like an exponential of a second degree polynomial in the spatial variable. In the second one, we prove a similar behaviour to the one corresponding to the wave equation. In these two sections we use in a relevant way an exponentially weighted Poincaré inequality which has been revealed very useful in several thermal and mechanical problems. This kind of results have relevance to understand the propagation of perturbations for nerve models.