# Entire solutions to reaction diffusion equations

## TL;DR

This paper investigates the stability and asymptotic behavior of entire solutions to reaction diffusion equations, providing new proofs and extending stability analysis to monostable cases using super-sub solution methods.

## Contribution

It offers the first analysis of local asymptotic stability for entire solutions in monostable reaction diffusion equations and simplifies existing proofs for bistable cases.

## Key findings

- Proved local exponential asymptotic stability for entire solutions in bistable and monostable cases.
- Extended stability analysis to monostable equations for the first time.
- Analyzed the asymptotic behavior of solutions as time approaches infinity.

## Abstract

In this paper, we first use the super-sub solution method to prove the local exponential asymptotic stability of some entire solutions to reaction diffusion equations, including the bistable and monostable cases. In the bistable case, we not only obtain the similar asymptotic stability result given by Yagisita in 2003, but also simplify his proof. For the monostable case, it is the first time to discuss the local asymptotic stability of entire solutions. Next, we will discuss the asymptotic behavior of entire solutions of bistable equations as $t\rightarrow+\infty$, since the other side was completely known. Here, our results are obtained by use of the asymptotic stability of constant solutions and pairs of diverging traveling front solutions of these equations, instead of constructing the corresponding super-sub solutions as usual.