Transition semi-wave solutions of reaction diffusion equations with free boundaries.

In this paper, we define the transition semi-wave solution (cf. Definition 1.1) of the following reaction diffusion equation with free boundaries (1) { u t = u x x + f (t , x , u) , t ∈ R , x < h (t) , u (t , h (t)) = 0 , t ∈ R , h ′ (t) = − μ u x (t , h (t)) , t ∈ R. In the homogeneous case, i.e., f (t , x , u) = f (u) , under the hypothesis f (u) ∈ C 1 ([ 0 , 1 ]) , f (0) = f (1) = 0 , f ′ (1) < 0 , f (u) < 0 for u > 1 , we prove that the semi-wave connecting 1 and 0 of (1) is unique provided it exists. Furthermore, we prove that any bounded transition semi-wave connecting 1 and 0 is exactly the semi-wave. In the cases where f is KPP-Fisher type and almost periodic in time (space), i.e., f (t , x , u) = u (c (t) − u) (resp. u (a (x) − u)) with c (t) (resp. a (x)) being almost periodic, applying totally different method, we also prove any bounded transition semi-wave connecting the unique almost periodic positive solution of u t = u (c (t) − u) (resp. u x x + u (a (x) − u) = 0) and 0 is exactly the unique almost periodic semi-wave of (1). Finally, we provide an example of the heterogeneous equation to show the existence of the transition semi-wave without any global mean speeds. [ABSTRACT FROM AUTHOR]