Sharp transition between extinction and propagation of reaction

Abstract

We consider the reaction-diffusion equation \[ T t = T x x + f ( T ) T_t = T_{xx} + f(T) \] on R {\mathbb {R}} with T 0 ( x ) ≡ χ [ − L , L ] ( x ) T_0(x) \equiv \chi _{[-L,L]} (x) and f ( 0 ) = f ( 1 ) = 0 f(0)=f(1)=0 . In 1964 Kanel ′ ^{\prime } proved that if f f is an ignition non-linearity, then T → 0 T\to 0 as t → ∞ t\to \infty when L > L 0 L>L_0 , and T → 1 T\to 1 when L > L 1 L>L_1 . We answer the open question of the relation of L 0 L_0 and L 1 L_1 by showing that L 0 = L 1 L_0=L_1 . We also determine the large time limit of T T in the critical case L = L 0 L=L_0 , thus providing the phase portrait for the above PDE with respect to a 1-parameter family of initial data. Analogous results for combustion and bistable non-linearities are proved as well.