Spectral comparison and gradient-like property in the FitzHugh–Nagumo type equations

In the study of stationary problems of the FitzHugh–Nagumo activator–inhibitor system, based on a variational structure, the associated Euler–Lagrange equation is an elliptic equation with a non-local term. We are concerned with the influence of such a variational structure on the local or global dynamics of the system. With the aid of spectral comparison, it will be shown that the dimension of the unstable manifold of an equilibrium solution to the system can actually be determined by the Morse index of the corresponding critical point. Also, some estimates for the eigenvalues concerned with the linearized stability are established. Besides the local structure near the equilibrium solutions, the existence of a Lyapunov function leads to a gradient-like property in the global dynamics. The case under consideration is when, in the equation for the inhibitor, the rate of degradation is relatively higher than that of production due to the activator. (paper)