Existence and uniqueness of entire solutions for a reaction–diffusion equation

We consider entire solutions of ut=uxx-f(u), i.e. solutions that exist for all , where f(0)=f(1)=00, we show that such entire solution exists and is unique up to space–time translations. In the case f′(1)<0, we derive two families of such entire solutions. In the first family, one cannot be any space–time translation of the other. Yet all entire solutions in the second family only differ by a space–time translation.