Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations

We study entire solutions of a scalar reaction-diffusion equation of 1-space dimension. Here the entire solutions are meant by solutions defined for all $(x,t)\in\mathbb R^2$. Assuming that the equation has traveling front solutions and using the comparison argument, we prove the existence of entire solutions which behave as two fronts coming from the both sides of $x$-axis. A key idea for the proof of the main results is to characterize the asymptotic behavior of the solutions as $t\to-\infty$ in terms of appropriate subsolutions and supersolutions. This argument can apply not only to a general bistable reaction-diffusion equation but also to the Fisher-KPP equation. We also extend our argument to the Fisher-KPP equation with discrete diffusion.