# Entire Solution in an Ignition Nonlocal Dispersal Equation: Asymmetric Kernel

## TL;DR

This paper investigates the existence and properties of entire solutions in a nonlocal dispersal equation with an asymmetric kernel, revealing how asymmetry influences wave profiles and speeds, and developing new methods for asymptotic analysis.

## Contribution

It introduces a novel approach to analyze asymptotic behaviors of traveling waves in asymmetric kernels and constructs new entire solutions with qualitative properties.

## Key findings

- Asymmetry of the kernel affects wave profiles and speeds.
- New methods are developed for asymptotic analysis without Ikehara theorem.
- Constructed entire solutions exhibit diverse qualitative behaviors.

## Abstract

This paper mainly focus on the front-like entire solution of a classical nonlocal dispersal equation with ignition nonlinearity. Especially, the dispersal kernel function $J$ may not be symmetric here. The asymmetry of $J$ has a great influence on the profile of the traveling waves and the sign of the wave speeds, which further makes the properties of the entire solution more diverse. We first investigate the asymptotic behavior of the traveling wave solutions since it plays an essential role in obtaining the front-like entire solution. Due to the impact of $f'(0)=0$, we can no longer use the common method which mainly depending on Ikehara theorem and bilateral Laplace transform to study the asymptotic rates of the nondecreasing traveling wave and the nonincreasing one tending to 0, respectively, thus we adopt another method to investigate them. Afterwards, we establish a new entire solution and obtain its qualitative properties by constructing proper supersolution and subsolution and by classifying the sign and size of the wave speeds.