Global existence and asymptotics of solutions of the Cahn–Hilliard equation

Abstract

This paper is concerned with the Cauchy problem of the Cahn–Hilliard equation { ∂ u ∂ t + Δ φ ( u ) + Δ 2 u = 0 , x ∈ R N , t > 0 , u | t = 0 = u 0 ( x ) , x ∈ R N . First, we construct a local smooth solution u ( t , x ) to the above Cauchy problem, then by combining some a priori estimates, Sobolev's embedding theorem and the continuity argument, the local smooth solution u ( t , x ) is extended step by step to all t > 0 provided that the smooth nonlinear function φ ( u ) satisfies a certain local growth condition at some fixed point u ¯ ∈ R and that ‖ u 0 ( x ) − u ¯ ‖ L 1 ( R N ) is suitably small. Secondly, we show that the global smooth solution u ( t , x ) satisfies the following temporal decay estimates: ‖ D k ( u ( t , x ) − u ¯ ) ‖ L p ( R N ) ⩽ c ( τ ) ( 1 + t ) − k 4 − N 4 ( 1 − 1 p ) , t ⩾ τ > 0 , k = 0 , 1 , … . Here p ∈ [ 1 , ∞ ] , c ( τ ) > 0 is a constant depending on τ and τ > 0 is any positive constant which can be chosen sufficiently small. At last, we show that, under a strong assumption on the growth of the nonlinear function φ ( u ) at u = u ¯ , the asymptotics of solutions of the above Cauchy problem is described by u ¯ + δ 0 t − N 4 G ( x t 4 ) . Here δ 0 = ∫ R N ( u 0 ( x ) − u ¯ ) d x , G ( x ) = ∫ R N exp ( − | η | 4 + i x ⋅ η ) d η .