Optimal $L^1$-type relaxation rates for the Cahn-Hilliard equation on the line

TL;DR

This paper establishes optimal algebraic relaxation rates for the Cahn-Hilliard equation on the line, showing how solutions approach a kink profile over time under certain initial conditions.

Contribution

It provides the first derivation of optimal relaxation rates for the Cahn-Hilliard equation towards kink solutions on the real line.

Findings

Derived optimal algebraic decay rates for the solution

Quantified decay of energy and perturbation over time

Used Nash inequalities, duality, and Schauder estimates

Abstract

In this paper we derive optimal algebraic-in-time relaxation rates to the kink for the Cahn-Hilliard equation on the line. We assume that the initial data have a finite distance---in terms of either a first moment or the excess mass---to a kink profile and capture the decay rate of the energy and the perturbation. Our tools include Nash-type inequalities, duality arguments, and Schauder estimates.