Travelling Fronts in Asymmetric Nonlocal Reaction Diffusion Equations: the Bistable and Ignition Cases

This paper is devoted to the study of the travelling front solutions which appear in a nonlocal reaction-diffusion equations of the form ∂u ∂t = J ⋆ u− u+ f(u). When the nonlinearity f is of bistable or ignition type, and the dispersion kernel J is asymmetric, the existence of a travelling wave is proved. The uniqueness of the speed of the front is also established. The construction of the front essentially relies on the vanishing viscosity techniques, some a priori estimates on the speed’s front and various comparison principles.