On the 2D Cahn-Hilliard/Allen-Cahn equation with the inertial term

Abstract

In this paper, we consider the initial boundary value problem for the hyperbolic relaxation of the 2D Cahn-Hilliard/Allen-Cahn equation with cubic nonlinearity of class $C^2$. We prove that the semigroup generated by the weak solutions of this problem possesses a global attractor. We also improve the previously obtained results concerning the global attractors for the 2D Cahn-Hilliard equation with the inertial term.

Introduction

The Cahn-Hilliard/Allen-Cahn equation

$$u_t=-\mathrm{\Delta }(\mathrm{\Delta }u-f\left(u\right))+(\mathrm{\Delta }u-f\left(u\right))$$

was introduced in [15], as a simplification of a mesoscopic model for multiple microscopic mechanism in surface processes, where f is the derivative of a nonconvex potential (e.g., $f\left(u\right)=u(u^2-1)$). The existence of the weak solutions of (1.1) on $\mathrm{\Omega }\times [0,T)$ with boundary conditions

$$\frac{\partial u}{\partial \nu }{|}_{\partial \mathrm{\Omega }}=\frac{\partial \mathrm{\Delta }u}{\partial \nu }{|}_{\partial \mathrm{\Omega }}=0$$

was studied in [16], where Ω is a smooth bounded domain in ${\mathbb{R}}^n$ ($n\le 3$) and ν is the unit normal on ∂Ω. When the term $(\mathrm{\Delta }u-f\left(u\right))$ is absent, then (1.1) becomes the well-known Cahn-Hilliard equation

$$u_t-\mathrm{\Delta }(-\mathrm{\Delta }u+f\left(u\right))=0,$$

which was introduced in [4] to describe the phase separation process in binary mixture. The mathematical aspects of (1.3), such as well-posedness, asymptotic behavior of solutions and existence of the attractors, have been widely investigated by many authors (see [2], [6], [7], [17], [20], [21], [22], [24], [25] and references therein). Although the Cahn-Hilliard equation shows, in general, a good agreement with experimental data of the binary systems, it has been noted that, in certain materials like glasses, the equation needs to be modified in order to describe non-equilibrium decomposition caused by the deep supercooling. In this respect, P. Galenko et al. (see [8] and [9]) have proposed hyperbolic relaxation of Cahn-Hilliard equation. This modified Cahn-Hillard equation is written as

$$\epsilon u_{tt}+u_t-\mathrm{\Delta }(-\mathrm{\Delta }u+f\left(u\right))=0,$$

where $\epsilon >0$ is a relaxation time.

The well-posedness and long-time dynamics of (1.4) were studied in [10], [11], [27], [28] in one dimensional case by taking the advantage of Sobolev embedding $H^1\hookrightarrow L^{\infty }$. The situation becomes difficult in higher dimensions due to the lack of this embedding. In [12], the attractors for the semigroups generated by (1.4) were studied in three dimensional case, for very small ε. In [13], the equation

$$u_{tt}+u_t-\mathrm{\Delta }(-\mathrm{\Delta }u+f\left(u\right))=g$$

was considered on $\mathrm{\Omega }\times [0,\infty )$, with boundary conditions

$$u{|}_{\partial \mathrm{\Omega }}=\mathrm{\Delta }u{|}_{\partial \mathrm{\Omega }}=0,$$

where g is a time-independent function and Ω is a smooth bounded domain in ${\mathbb{R}}^2$. The authors of that paper reduced (1.5)-(1.6) to the abstract differential equation

$$u_{tt}+u_t+A(Au+f\left(u\right))=g,$$

and assuming

$$\{\begin{array}{c}f\in C_{loc}^{2,1}\left(\mathbb{R}\right),f(0)=0,\left|f^{″}\left(r\right)\right|\le C(1+\left|r\right|),\forall r\in \mathbb{R},\hfill \\ \exists \lambda \in [0,\infty ):f^{\prime }\left(r\right)\ge -\lambda ,\forall r\in \mathbb{R},\hfill \\ \underset{\left|r\right|\to \infty }{\lim \inf }\frac{f(r)}{r}>-{\lambda }_1,\hfill \end{array}$$

they proved the existence of the global attractors for the quasi-strong solutions in $H^3\left(\mathrm{\Omega }\right)\times H^1\left(\mathrm{\Omega }\right)$ and for the energy bounded solutions in $H_0^1\left(\mathrm{\Omega }\right)\times H^{-1}\left(\mathrm{\Omega }\right)$, where A= −Δ with $D(A)=H^2\left(\mathrm{\Omega }\right)\cap H_0^1\left(\mathrm{\Omega }\right)$ and ${\lambda }_1$ is the first eigenvalue of A. However, the existence of the global attractor for the weak solutions was left as an open question due to the lack of dissipativity (see [13, Remark 6.8]). Later, in [14], the authors developed the exponential regularization method and assuming, in addition to (1.8), the conditions

$$f\in C^4\left(\mathbb{R}\right)\text{ and }{f}^{‴}\in L^{\infty }\left(\mathbb{R}\right)$$

they established the dissipativity (see [14, Theorem 7.4]), which leads to the existence of the global attractor for the weak solutions of (1.7). Recently, in [18], the conditions (1.9) have been removed for sub-cubic nonlinearities. However, the existence of the global attractor for the weak solutions of (1.7) with the cubic nonlinearity, without assuming the additional regularity conditions (1.9), has remained an open question.

In this paper, we deal with the hyperbolic relaxation of (1.1) with the boundary conditions (1.2). Namely, we consider the initial boundary value problem

$$\{\begin{array}{c}{u}_{tt}+u_t-\mathrm{\Delta }(-\mathrm{\Delta }u+f\left(u\right))+(-\mathrm{\Delta }u+f\left(u\right))=0,(t,x)\in (0,\infty )\times \mathrm{\Omega },\hfill \\ \frac{\partial u}{\partial \nu }=\frac{\partial \mathrm{\Delta }u}{\partial \nu }=0,(t,x)\in (0,\infty )\times \partial \mathrm{\Omega },\hfill \\ u(0,x)=u_0\left(x\right),u_t(0,x)=u_1\left(x\right),x\in \mathrm{\Omega },\hfill \end{array}$$

where $\mathrm{\Omega }\subset {\mathbb{R}}^2$ is a smooth bounded domain and ν is the unit normal on ∂Ω. The equation (1.10)₁, with the boundary conditions (1.10)₂, can be written in the form of (1.7), with the operator $A=-\mathrm{\Delta }+I$, $D(A)=\{u\in H^2\left(\mathrm{\Omega }\right):\frac{\partial u}{\partial \nu }{|}_{\partial \mathrm{\Omega }}=0\}$ and with the nonlinearity $f(u)-u$ instead of $f(u)$. In this case, since ${\lambda }_1=1$, the condition corresponding to (1.8)₃ is

$$\underset{\left|r\right|\to \infty }{\lim \inf }\frac{f(r)}{r}>0.$$

So, assuming the conditions (1.8)₁, (1.8)₂, (1.9) and (1.11), one can apply the method of [14] to (1.10).

Here, we remove the regularity conditions (1.9) and the monotonicity condition (1.8)₂, and replace (1.11) by a weaker condition (see (2.3) below). We also give a positive answer to the question raised in [13, Remark 6.8], without assuming any additional condition on the nonlinearity, and thereby improve the result obtained in [14], concerning the dissipativity of the weak solutions of 2D Cahn-Hilliard equation with the inertial term (see Remark 3.1). Moreover, we improve the result obtained in [13], concerning the existence of the global attractor for the quasi-strong solutions. Namely, we prove the existence of the global attractor for the quasi-strong solutions of 2D Cahn-Hilliard equation with the inertial term, when $f^{″}\left(s\right)$ grows at infinity as ${\left|s\right|}^{\frac{3}{2}}$ (see Remark 3.2).