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\rightarrow \infty , of solutions of \begin{cases} u_{t} = u_{xx} + f(u), & x > 0,\:t > 0, \\ u(0,t) = bu_{x}(0,t), & t > 0, \\ u(x,0) = u_{0}(x)⩾0, & x⩾0, \end{cases} where b⩾0 and f is an unbalanced bistable nonlinearity. By investigating families of initial data of the type \{\sigma \phi \}_{\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\mathrm{\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.