Title: Entire Solution in an Ignition Nonlocal Dispersal Equation: Asymmetric Kernel,
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.