We study the existence and asymptotic stability of traveling waves to

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We consider entire solutions of u t = u xx − f (u), i.e. solutions that exist for all (x, t) ∈ R 2 , where f (0) = f (1) = 0 < f (0). In particular, we are interested in the entire solutions which behave as two opposite wave fronts of positive speed(s) approaching each other from both sides of the x-axis and then annihilating in a finite time. In the case f (1) > 0, 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.

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