Open Access
Issue
Wuhan Univ. J. Nat. Sci.
Volume 27, Number 5, October 2022
Page(s) 361 - 366
DOI https://doi.org/10.1051/wujns/2022275361
Published online 11 November 2022

© Wuhan University 2022

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

This paper is devoted to pointwise estimate of following Cahn-Hilliard equation with inertial term:

(1)

Here is the usual Laplace operator, is a given constant. The nonlinear term has the form or , where , are positive integers and

Equation (1) is closely related to the well-known Cahn-Hilliard system

(2)

System (2) is a hyperbolic equation with relaxation which describes phase separation of a binary mixture and denotes the relative concentration of one phase. The fourth order differential operator of (2) makes its mathematical analysis more difficult than the corresponding second order equation[1]. Due to the physical background and mathematical difficulties, many mathematicians devoted their enthusiasm to the equation and got much qualitative behavior of the solution (see e.g. Refs.[2-8]). In order to model non-equilibrium decompositions caused by deep supercooling in certain glasses, Galenko et al[9] advised to append inertial term to (2). The unknown reflects the relative concentration of one phase. The modified system (1) shows a good agreement with experiments performed on glasses[9,10]. For simplicity, we later suppose .

The mathematical structure changed after the adjunction. Equation (1) is a hyperbolic equation with relaxation while (2) is a parabolic one, so they present different mathematical features. Eq. (1) has some mathematical difficulties because there is no regularization of the solution in finite time anymore. In order to get regularization, mathematicians often first study them with viscous term. Xu and Shi[11] got global existence with large initial data for any space dimension. Because of weak dissipation, previous work for (1) mainly focused on the so-called energy bounded solution and quasi-strong solution[12-14]. Wang and Wu[15] took advantage of frequency decomposition and energy method, and they got global existence and decay rate of classical solution of (1) for the case of with small initial data. Based on their work, Li and Mi[16] got pointwise decay estimate of the solution for . Their decay rates are closely related to the space dimension . The solution decays faster if is larger which makes it much more difficult to deal with lower space dimension. Obviously, compared with reflects the reality. We make much more delicate analysis of the nonlinear term with convolution of the Green function, and get the same decay rate as those of Refs.[15, 16].

We introduce some notations in this paper . We denote or a constant or constant depending on variable . denote usual Lebesgue and Sobolev spaces on and , with norms , , , respectively. is a multi-index with . Fourier transform to the variable of function is , that is , where is the imaginary unit. Thus the inverse Fourier transform to the variable of is defined as

The rest of this paper is arranged as follows. In Section 1, we give some preparing work. The estimate of the solution will be given in Section 2.

1 Preliminary Work

Our work is a follow-through of the global existence of (1), that is the Theorem 1.1 of Ref.[15]. We list it here.

Theorem 1[15] If initial data satisfy

for some small , , the Cauchy problem (1) admits a unique, global, classical solution satisfying:

The Green function of (1) is defined as

Then where

Using Duhamel principle, the solution of (1) can be represented as

(3)

Operator denotes the convolution of space variable x in this paper.

We will use frequency decomposition to estimate . Set

are smooth cut-off functions. Here . Set .

Denote

For , we can use the results of Ref.[16] which are Proposition 3.1 and Proposition 3.2. We list them here.

Theorem 2[16] There exists positive constant , such that

Since using the same method of Theorem 2, we can get the same conclusion, that is

Theorem 3[16] There exists positive constant, such that

When is large enough, using Taylor expansion, we get

(4)

(5)

In order to get estimate of high frequency part, we need to understand the construction of , and we can use Lemma 2.5 in Ref.[17]. That is

Lemma 1[17] Assume that with

then there exist distributions , and a constant such that

where is the Dirac function. Furthermore, for a positive integer ,

with being sufficiently small.

From (4), (5) and Lemma 1, we have the following construction

(6)

where satisfy Lemma 1.

2 Decay Estimation

We use (3) and decay rate of to estimate the solution step by step.

Theorem 4   If , satisfy the condition of Theorem 1, and

we have .

Proof   Here and afterwards we take . We divide the following integral into three parts.

(7)

If , we have

(8)

Thus

(9)

When , we have , then

(10)

From (7), (9), (10) and Theorem 2, we have

(11)

Suppose function , from (4), (5), we have

(12)

Take , we have

(13)

If , take , we have

(14)

From (13), (14), we have

(15)

We know , , from (15), we get

(16)

From (11), (16), the theorem is proved.

Using the same method as that of Theorem 4, we also get the following theorem.

Theorem 5   If , we have

Next, we begin to estimate the nonlinear term .

Set , . Then

(17)

From (17), for , noticing , we have

(18)

Theorem 6   For , we have

Proof   We first divide the following integral into four items.

(19)

From (8), we have

Noticing , from (8), we have

From (19) and above four inequalities, we get

(20)

From (20), (18), Theorem 2, we have

(21)

If , from (20), we get

(22)

If with small enough, we have

Thus

(23)

(24)

From (22), (23), (24), (18), (6), we get

(25)

From (21), (25), the theorem is proved.

From (3), Theorem 4, Theorem 5, and Theorem 6, we get

From the definition of , we get

Because , are small enough, thus is bounded. Then we have

Theorem 7   is the main conclusion of this paper.

Theorem 7   If satisfy with small enough, , with , then the Cauchy problem of (1) exists a global classical solution , furthermore .

Remark 1   From (8), our result coincides with and decay rate of Refs.[14,15]. Furthermore, we move forward the solution's derivative order which can be estimated by module.

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