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 (x,t)∈R2, 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.