Pointwise Estimates of Solutions for the Viscous Cahn-Hilliard Equation with Inertial Term

In this paper, we study the pointwise estimates of solutions to the viscous Cahn-Hilliard equation with the inertial term in multidimensions. We use Green’s function method. Our approach is based on a detailed analysis on the Green’s function of the linear system. And we get the solution’s Lp convergence rate.


Introduction
In this paper, we study the pointwise estimates of the solution ρðx, tÞ to the Cauchy problem: where n ≥ 4. f ðρÞ is the intrinsic chemical potential which is smooth in the small neighborhood of the origin, and f ðρÞ = Oðρ 1+θ Þ when jρj ≤ 1 and θ is a positive integer. When η = 0, Eq. (1) is the well-known Cahn-Hilliard equation. When η ≠ 0, ηρ tt is the inertial term. When k ≠ 0, −kΔρ t is the viscous term. Without loss of generality, we let η = 1 and k = 1.
The classical Cahn-Hilliard equation was proposed in the sixties by Cahn and Hilliard which describes the phase separation in materials science, and it has been widely studied. The reader may see references ( [1][2][3][4][5][6]) and the related references therein. The Cahn-Hilliard equations with inertial term model nonequilibrium decompositions caused by deep supercooling in certain glasses. As we know, the wellknown Cahn-Hilliard equation is a parabolic equation, but the Cahn-Hilliard equation with the inertial term is a hyper-bolic equation with relaxation which brings many mathematical difficulties to study. For which, without smallness assumption on initial data, [7] got the global existence of the classical solution. [8] obtained the global existence and the optimal decay rate of the classical solution by the Fourier splitting method. Wang and Wu [9] obtained the global existence and optimal decay rate of the classical solution by long wave-short wave method. Li and Mi [10] got the pointwise estimates and the L p (1 ≤ p≤∞) convergence rate of the solution by Green's function method. Some other works on the Cahn-Hilliard equation with the inertial term can be seen in [11][12][13].
For viscous Cahn-Hilliard equation, [14] discussed the large time behavior of solutions when the dimension n ≤ 5. For the viscous Cahn-Hilliard equation with the inertial term, it describes the early stages of spinodal decomposition in certain glasses (see [15][16]). And for which, [17] established the existence of families of exponential attractors and inertial manifolds; [18] studied the long time dynamic of the system in three-dimensional. In this paper, we are interested in the viscous Cahn-Hilliard equation with the inertial term. Under the smallness assumption on initial data, based on the detailed analysis of the Green's function, we get the pointwise estimates of solutions. From the representation of the symbol value to the Green's function for the linear problem of Eq. (1), we also find that the decay rate mainly depends on the lower-frequency part, i.e., the long wave part.
It is shown that the solution's decay rate is the same as [10]. Our study bases on Section 4 in [9].
To the best of our knowledge, this is the first time to obtain the pointwise estimates of the solution to Eq. (1).
Throughout this paper, C denotes the generic positive constants. W m,p = W m,p ðℝ n Þ ðm ∈ ℤ + , p ∈ ½1,∞Þ denote the usual Lebesgue space with norms k·k L p and the usual Sobolev space with its norm In particular, we use W m,2 = H m . The main result can be stated as following Theorem 1: then for |β | <n, the solution to Eq. (1) has the following estimates: where ε and E is sufficiently small positive constants, ½m = max fa | a ∈ ℤ, a ≤ mg, B d ðjxj, tÞ = ð1 + ðjxj 2 /1 + tÞÞ −d : Under the assumptions of Theorem 1, for p ≥ 1, jβj < n, we have that Remark 3. We get the same decay rate of the solution as [10].

Remark 4.
Our study bases on [9,10], where the spacial dimension n ≥ 4. Then in this paper, we have the same assumptions for the spacial dimension.

Proposition 7.
For fixed s and R, there exist positive numbers m and C such that The proof of Proposition 7 is similar to that of Proposition 12 in [10], so we omit it.
Then, we havê