An entire solution of a bistable parabolic equation on R with two colliding pulses

We consider semilinear parabolic equations of the form ut = uxx + f(u), x ∈ R, t ∈ I, (A) 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). ∗Supported in part by the NSF Grant DMS–1161923