ISON A SIXTH-ORDER CAHN-HILLIARD SYSTEM WITH TEMPERATURE

Abstract

Our aim in this paper is to study the asymptotic behavior, in terms of finite-dimensional attractors, of the sixth-order Cahn-Hilliard system with temperature, namely,

$$
\left\{\begin{array}{l}
\frac{\partial \varphi}{\partial t}=\Delta \mu \\
\mu=f(\varphi)-\Delta \varphi+\pi f^{\prime}(\varphi)-\Delta \pi-\theta \\
\pi=f(\varphi)-\Delta \varphi \\
\frac{\partial \theta}{\partial t}-\Delta \theta=-\frac{\partial \varphi}{\partial t}
\end{array}\right.
$$

in $\Omega \times(0,+\infty)$, subject to the Dirichlet boundary conditions (for simplicity):

$$
\varphi=\mu=\pi=\theta=0 \text { on } \Gamma \times(0,+\infty)
$$

and the initial conditions

$$
\left.\varphi\right|_{t=0}=\varphi_0,\left.\theta\right|_{t=0}=\theta_0 \text { in } \Omega.
$$

In this context, $\varphi$ is the order parameter, $\theta$ is the relative temperature (defined as $\theta=\widetilde{\theta}-\theta_E$, where $\widetilde{\theta}$ is the absolute temperature and $\theta_E$ is the equilibrium melting temperature), $\Omega$ is the domain occupied by the system (we assume here that it is a smooth and bounded domain $\mathbb{R}^n, n=1,2,3$ ), and $f$ is the derivative of a double-well potential (a typical choice of the potential is $F(s)=\frac{1}{4}\left(s^2-1\right)^2$, hence the usual cubic nonlinear term $\left.f(s)=s^3-s\right).$

This system is based on a modification of the Ginzburg-Landau free energy proposed in [1] (see also [9]). In particular, the free energy contains an additional term called Willmore regularization. We prove the existence, uniqueness and regularity of solutions, as well as the existence of the global attractor.

Received: March 21, 2024
Accepted: May 10, 2024