Long Time Behavior of Entire Solutions to Bistable Reaction Diffusion Equations

Abstract

The first aim in the paper is to prove the local exponential asymptotic stability of some entire solutions to bistable reaction diffusion equations via the super-sub solution method. If the integral of the reaction term $f$ over the interval $[0,1]$ is positive, we not only obtain the similar asymptotic stability result found by Yagisita in 2003, but also simplify the proof. The asymptotic stability result for the case $\int^1_{0} f(u) \, du \lt 0$ is also obtained, which is not considered by Yagisita. After that, the asymptotic behavior of entire solutions as $t \to +\infty$ is investigated, since the other side was completely known. Here, the result is established by use of the asymptotic stability of constant solutions and pairs of diverging traveling front solutions, instead of constructing the super-sub solutions as usual. Finally, for the special bistable case $f(u) = u(1-u)(u-\alpha)$, $\alpha \in (0,1)$, we prove the entire solution continuously depends on $\alpha$.