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

### Xinfu Chen

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA### Bendong Lou

Department of Mathematics, Tongji University, Shanghai 200092, China### Maolin Zhou

Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo 153-8914, Japan### Thomas Giletti

Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo 153-8914, Japan, Institut Elie Cartan de Lorraine, Université de Lorraine, Vandoeuvre-lès-Nancy 54506, France

## Abstract

We study the long time behavior, as $t→∞$, of solutions of

where $b⩾0$ and *f* is an unbalanced bistable nonlinearity. By investigating families of initial data of the type ${σϕ}_{σ>0}$, 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 $σ_{⁎}$ such that the solution converges uniformly to 0 for any $0<σ<σ_{⁎}$, and locally uniformly to a positive stationary state for any $σ>σ_{⁎}$. In the threshold case $σ=σ_{⁎}$, 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 $Clnt$ 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.

## Cite this article

Xinfu Chen, Bendong Lou, Maolin Zhou, Thomas Giletti, Long time behavior of solutions of a reaction–diffusion equation on unbounded intervals with Robin boundary conditions. Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 1, pp. 67–92

DOI 10.1016/J.ANIHPC.2014.08.004