An entire solution of a bistable parabolic equation on [formula omitted] with two colliding pulses

Abstract

We consider semilinear parabolic equations of the form(A)ut=uxx+f(u),x∈R,t∈I, where I=(0,∞) or I=(−∞,∞). Solutions defined for all (x,t)∈R2 are referred to as entire solutions. Assuming that f∈C1(R) is of a bistable type with stable constant steady states 0 and γ>0, we show the existence of an entire solution U(x,t) of the following form. For t≈−∞, U(⋅,t) has two humps, or, pulses, one near ∞, the other near −∞. As t increases, the humps move toward the origin x=0, eventually “colliding” and forming a one-hump final shape of the solution. With respect to the locally uniform convergence, the solution U(⋅,t) is a heteroclinic orbit connecting the (stable) steady state 0 to the (unstable) ground state of the equation uxx+f(u)=0. We find the solution U as the limit of a sequence of threshold solutions of the Cauchy problem for equation (A).