Asymptotic Profile of Solutions to a Hyperbolic Cahn-Hilliard Equation
by Hiroshi Takeda   Yasunori Maekawa   Shuichi Kawashima  

Vol. 10 No. 4 (2015) P.479~P.539

ABSTRACT

We study the initial value problem for a hyperbolic Cahn-Hilliard equation in n-dimensional space. The dissipative structure of our linearized equation is of the regularity-loss type. We overcome the difficulty caused by the regularity-loss property by introducing a set of suitable time-weighted spaces and prove the global existence and optimal decay of solutions under smallness and enough regularity assumptions on the initial data. Moreover, we investigate the asymptotic behavior of our nonlinear solutions as t → $\infty$. When n $\geq$ 3, they are asymptotic to the linear diffusion wave expressed by the fundamental solution of the equation $v_{t}$ +$\Delta ^{2}v$ = 0. On the other hand, when n = 1 or n = 2, they are asymptotic to the nonlinear diffusion wave which can be expressed in terms of the self-similar solution of the equation $v_{t}$ + $\Delta ^{2}v$ = $\Delta$ $v^{1+\frac{2}{n}}$.

KEYWORDS
hyperbolic Cahn-Hilliard equation; regularity-loss property, global existence; decay estimate; asymptotic behavior.

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 35L30, 35L76, 35B40, 35C06.

MILESTONES

Received: 2015-03-23
Revised : 2015-06-08
Accepted: 2015-06-08

Download Full Content