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

This paper is concerned with the Cauchy problem of the Cahn–Hilliard equation{∂u∂t+Δφ(u)+Δ2u=0,x∈RN,t>0,u|t=0=u0(x),x∈RN. First, we construct a local smooth solution u(t,x)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)u(t,x) is extended step by step to all t>0t>0 provided that the smooth nonlinear function φ(u)φ(u) satisfies a certain local growth condition at some fixed point u¯∈R and that ‖u0(x)−u¯‖L1(RN) is suitably small. Secondly, we show that the global smooth solution u(t,x)u(t,x) satisfies the following temporal decay estimates:‖Dk(u(t,x)−u¯)‖Lp(RN)⩽c(τ)(1+t)−k4−N4(1−1p),t⩾τ>0,k=0,1,…. Here p∈[1,∞]p∈[1,∞], c(τ)>0c(τ)>0 is a constant depending on τ   and τ>0τ>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)φ(u) at u=u¯, the asymptotics of solutions of the above Cauchy problem is described by u¯+δ0t−N4G(xt4). Here δ0=∫RN(u0(x)−u¯)dx, G(x)=∫RNexp(−|η|4+ix⋅η)dη.