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

This paper is concerned with the Cauchy problem of the Cahn–Hilliard equation First, we construct a local smooth solution u(t,x) to the above Cauchy problem, then by combining some a priori estimates, Sobolevʹs 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 and that is suitably small. Secondly, we show that the global smooth solution u(t,x) satisfies the following temporal decay estimates: 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 , the asymptotics of solutions of the above Cauchy problem is described by . Here , .