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

Abstract

We study entire solutions of a scalar reaction-diffusion equationof 1-space dimension. Here the entire solutions are meant by solutionsdefined for all $(x,t)\in\mathbb R^2$.Assuming that the equation has traveling front solutions and using thecomparison argument, we prove the existence of entire solutions whichbehave as two fronts coming from the both sides of $x$-axis.A key idea for the proof of the main results is to characterize theasymptotic behavior of the solutions as $t\to-\infty$ in terms ofappropriate subsolutions and supersolutions. This argument can applynot only to a general bistable reaction-diffusion equation but also tothe Fisher-KPP equation.We also extend our argument to the Fisher-KPP equation withdiscrete diffusion.