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 ϕ 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 ${\sigma }^{⁎}$ such that the solution converges uniformly to 0 for any $0<\sigma <{\sigma }^{⁎}$, and locally uniformly to a positive stationary state for any $\sigma >{\sigma }^{⁎}$. In the threshold case $\sigma ={\sigma }^{⁎}$, 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.