## Differential and Integral Equations

- Differential Integral Equations
- Volume 25, Number 3/4 (2012), 341-362.

### Asymptotic profile of solutions for the limit unstable Cahn-Hilliard equation with inertial term

Hiroshi Takeda and Shuji Yoshikawa

#### Abstract

The aim of this paper is to investigate the asymptotic behavior of solutions to the limit unstable Cahn-Hilliard equation with inertial term $$ \partial_t^2 u + \partial_t u + \Delta ( \Delta u + | u |^{p-1} u)=0, \quad {\rm in } \quad (0,\infty) \times {\mathbb{R}}^n. $$ We shall prove the unique global existence of a small solution for the equation in the super critical case $p > 1 + \frac{2}{n}$ in $n=1,2,3$, and also give an asymptotic profile of the solution. By considering the higher-order expansion of the solution, we obtain more precise information about optimal decay of the solution under the more restricted condition $p > 1+ \frac{4}{n}$, and observe the contribution of the nonlinear term to the solution.

#### Article information

**Source**

Differential Integral Equations, Volume 25, Number 3/4 (2012), 341-362.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356012738

**Mathematical Reviews number (MathSciNet)**

MR2917886

**Zentralblatt MATH identifier**

1265.35052

**Subjects**

Primary: 35G25: Initial value problems for nonlinear higher-order equations 35A01: Existence problems: global existence, local existence, non-existence 35B40: Asymptotic behavior of solutions 82C26: Dynamic and nonequilibrium phase transitions (general)

#### Citation

Takeda, Hiroshi; Yoshikawa, Shuji. Asymptotic profile of solutions for the limit unstable Cahn-Hilliard equation with inertial term. Differential Integral Equations 25 (2012), no. 3/4, 341--362. https://projecteuclid.org/euclid.die/1356012738