Traveling Wave Solutions of a Reaction-Diffusion Equation with State-Dependent Delay

TL;DR

This paper investigates traveling wave solutions in reaction-diffusion equations with state-dependent delays, establishing existence, nonexistence, and minimal wave speed under different conditions using fixed point methods.

Contribution

It provides new results on the existence and properties of traveling wave solutions for reaction-diffusion equations with state-dependent delays, including cases with non-monotone birth functions.

Findings

Existence of monotone traveling waves when the birth function is monotone.

Nonexistence of such waves under certain conditions.

Determination of minimal wave speed for non-monotone birth functions.

Abstract

This paper is concerned with the traveling wave solutions of a reaction-diffusion equation with state-dependent delay. When the birth function is monotone, the existence and nonexistence of monotone traveling wave solutions are established. When the birth function is not monotone, the minimal wave speed of nontrivial traveling wave solutions is obtained. The results are proved by the construction of upper and lower solutions and application of the fixed point theorem.