Traveling wave phenomena of a nonlocal reaction-diffusion equation with degenerate nonlinearity

Abstract

This paper deals with traveling wave phenomena of a degenerate reaction-diffusion equation with the nonlocal effect. We study the existence of traveling wave solutions which may be non-monotonic based on the two-point boundary value problem and Schauder’s fixed point theorem. We are excited to find that the unknown positive steady state is exactly a unique positive equilibrium for the large wave speed and the monotonicity of traveling waves depends on the wave speed. On the other hand, we demonstrate the existence of monotone traveling wave solutions by using the monotone iteration and operator theory, and investigate how wave profile (e.g. being non-monotone or periodic) is affected by the enhancement of nonlocality, parameter values and variation of initial values for two different forms of convolution kernel functions.