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In this paper, we first use the super-sub solution method to prove the localexponential asymptotic stability of some entire solutions to reaction diffusionequations, including the bistable and monostable cases. In the bistable case,we not only obtain the similar asymptotic stability result given by Yagisita in2003, but also simplify his proof. For the monostable case, it is the firsttime to discuss the local asymptotic stability of entire solutions. Next, wewill discuss the asymptotic behavior of entire solutions of bistable equationsas $t\rightarrow+\infty$, since the other side was completely known. Here, ourresults are obtained by use of the asymptotic stability of constant solutionsand pairs of diverging traveling front solutions of these equations, instead ofconstructing the corresponding super-sub solutions as usual.

Entire solutions to reaction diffusion equations