We study travelling fronts in a two-component singularly perturbed (FitzHugh-Nagumo-like) reaction-diffusion system with spatially heterogeneous coefficients. A front solution has a sharp interface and connects two equilibrium states (background states) of the PDE, but due to the spatial inhomogeneity these background states are not constant in space. This renders a comoving frame approach in order to construct travelling fronts practically useless. The difficulty is worsened by the fact that both the speed and the shape of the profile a front are varying in time, as clearly seen in numeric simulations.

To deal with the challenges, we develop new techniques to deal with the spatiotemporal nature of the problem. We manage to prove existence of travelling fronts for a large range of parameters. In doing so, we find numerous insights on the circumstances for an interface to persist in time. In addition, we find a satisfying description of how travelling fronts move. Specifically, we discover that the position of a travelling front can be described in the singular limit by means of a delay differential equation. Numerically this delay equation seems to provide a very precise approximation for the position of the front for small values of the singularly perturbed parameter.