Long time behavior of solutions of a reaction–diffusion equation on unbounded intervals with Robin boundary conditions

Abstract

We study the long time behavior, as $t\to \infty$, of solutions of

where $b⩾0$ and $f$ is an unbalanced bistable nonlinearity. By investigating families of initial data of the type $\{\sigma \varphi {\}}_{\sigma >0}$, where $\varphi$ belongs to an appropriate class of nonnegative compactly supported functions, we exhibit the sharp threshold between vanishing and spreading. More specifically, there exists some value ${\sigma }^{\ast }$ such that the solution converges uniformly to 0 for any $0<\sigma <{\sigma }^{\ast }$, and locally uniformly to a positive stationary state for any $\sigma >{\sigma }^{\ast }$. In the threshold case $\sigma ={\sigma }^{\ast }$, the profile of the solution approaches the symmetrically decreasing ground state with some shift, which may be either finite or infinite. In the latter case, the shift evolves as $C\ln t$ where $C$ is a positive constant we compute explicitly, so that the solution is traveling with a pulse-like shape albeit with an asymptotically zero speed. Depending on $b$, but also in some cases on the choice of the initial datum, we prove that one or both of the situations may happen.