Transition Fronts in Time Heterogeneous Bistable Equations with Nonlocal Dispersal: Existence, Regularity, Stability and Uniqueness

Abstract

This dissertation is devoted to the existence, regularity, stability and uniqueness of transition fronts in nonlocal bistable equations in time heterogeneous media.

Instead of existence, we start our study with qualitative properties. We first show that any transition front must be regular in space and propagates to the right with a continuously differentiable interface location function; this is also true in space-time heterogeneous media.

We then turn to the study of space nonincreasing transition fronts and prove various important properties such as uniform steepness, stability, uniform stability, exponential decaying estimates and so on.

Moreover, we show that any transition front, after certain space shift, coincides with a space nonincreasing transition front, verifying the uniqueness, up to space shifts, of transition fronts, and hence, all transition fronts satisfy just mentioned qualitative properties.

Also, we show that a transition front must be a periodic traveling wave in time periodic media and the asymptotic speeds of transition fronts exist in time uniquely ergodic media.

Finally, we prove the existence of space nonincreasing transition fronts by constructing appropriate approximating front-like solutions with regularities; this is done under certain additional assumptions.