Relaxation to Equilibrium in the One-Dimensional Cahn-Hilliard Equation

We study the stability of a so-called kink profile for the one-dimensional Cahn– Hilliard problem on the real line. We derive optimal bounds on the decay to equilibrium under the assumption that the initial energy is less than three times the energy of a kink and that the initial Ḣ−1 distance to a kink is bounded. Working with the Ḣ−1 distance is natural, since the equation is a gradient flow with respect to this metric. Indeed, our method is to establish and exploit elementary algebraic and differential relationships among three natural quantities: the energy, the dissipation, and the Ḣ−1 distance to a kink. Along the way it is necessary and possible to control the timedependent shift of the center of the L2 closest kink. Our result is different from earlier results because we do not assume smallness of the initial distance to a kink ; we assume only boundedness.