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 
Mathematics
CLC number: O 175.28
Pointwise Estimate of CahnHilliard Equation with Inertial Term in ℝ^{3}
College of Science, Hohai University, Nanjing 211100, Jiangsu, China
Received:
10
June
2022
Cauchy problem of CahnHilliard equation with inertial term in threedimensional space is considered. Using delicate analysis of its Green function and its convolution with nonlinear term, pointwise decay rate is obtained.
Key words: CahnHilliard equation with inertial term / Green function / pointwise decay
Biography: XU Hongmei, female, Ph.D., Associate professor, research direction: partial differential equations. Email: xxu_hongmei@163.com
Fundation item: Supported by the National Natural Science Foundation of China (11801137)
© Wuhan University 2022
This 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 CahnHilliard equation with inertial term:
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 wellknown CahnHilliard system
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.[28]). In order to model nonequilibrium 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 socalled energy bounded solution and quasistrong solution^{[1214]}. 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 multiindex 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 followthrough 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
Operator denotes the convolution of space variable x in this paper.
We will use frequency decomposition to estimate . Set
are smooth cutoff 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
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
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.
If , we have
Thus
When , we have , then
From (7), (9), (10) and Theorem 2, we have
Suppose function , from (4), (5), we have
Take , we have
If , take , we have
From (13), (14), we have
We know , , from (15), we get
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
From (17), for , noticing , we have
Theorem 6 For , we have
Proof We first divide the following integral into four items.
From (8), we have
Noticing , from (8), we have
From (19) and above four inequalities, we get
From (20), (18), Theorem 2, we have
If , from (20), we get
If with small enough, we have
Thus
From (22), (23), (24), (18), (6), we get
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.
References
 Liu S Q, Wang F, Zhao H J. Global existence and asymptotics of solutions of the CahnHilliard equation[J]. J Differential Equations, 2007, 238(2): 426469. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Caffarelli L A, Muler N E. An bound for solutions of the CahnHilliard equation[J]. Arch Ration Mech Anal, 1995, 133(2): 129144. [NASA ADS] [CrossRef] [Google Scholar]
 Cahn J W, Hilliard J E. Free energy of a nonuniform system[J]. I Interfacial Free Energy J Chem Phys, 1958, 28(2): 258267. [Google Scholar]
 Efendiev M, Miranville A, Zelik S. Exponential attractors for a singularly perturbed CahnHilliard system[J]. Math Nachr, 2004, 272(1): 1131. [CrossRef] [MathSciNet] [Google Scholar]
 Elliott C M, Zheng S. On the CahnHilliard equation[J]. Arch Ration Mech Anal, 1986, 96(4): 339357. [NASA ADS] [CrossRef] [Google Scholar]
 Gatti S, Grasselli M, Miranville A. On the hyperbolic relaxation of the onedimensional CahnHilliard equation[J]. J Math Anal Appl, 2005, 312(1): 230247. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Zheng S M, Milani A. Exponential attractors and inertial manifolds for singular perturbations of the CahnHilliard equations[J]. Nonlinear Anal, Theory Methods Appl, Ser A, 2004, 57 (5/6): 843877. [CrossRef] [MathSciNet] [Google Scholar]
 Zheng S M, Milani A. Global attractors for singular perturbations of the CahnHilliard equations[J]. J Differ Equations, 2005, 209(1): 101139. [NASA ADS] [CrossRef] [Google Scholar]
 Galenko P, Lebedev V. Noneqilibrium effects in spinodal decomposition of a binary System[J]. Phys Lett, 2008, 372 (7): 985989. [NASA ADS] [CrossRef] [Google Scholar]
 Grasselli M, Petzeltová H, Schimperna G. Asymptotic behavior of a nonisothermal viscous CahnHilliard equation with inertial term[J]. J Differ Equations, 2007, 239(1): 3860. [NASA ADS] [CrossRef] [Google Scholar]
 Xu H M, Shi Y F. Global existence and L^{p} decay estimate of solutions for viscous CahnHilliard equation with inertial term[J]. Wuhan University Journal of Natural Sciences, 2019, 24(6): 461466. [CrossRef] [Google Scholar]
 Grasselli M, Shilmperna G, Segatti A, et al. On the 3D CahnHilliard equation with inertial term[J]. J Evol Equ, 2009, 9(2): 371404. [CrossRef] [MathSciNet] [Google Scholar]
 Grasselli M, Shilmperna G, Zelik S. On the 2D CahnHilliard equation with inertial term[J]. Commun Partial Differ Equations, 2009, 34(2): 137170. [CrossRef] [Google Scholar]
 Grasselli M, Shilmperna G, Zelik S. Trajectory and smooth attractors for CahnHilliard equations with inertial term[J]. Nonlinearity, 2010, 23(3): 707737. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Wang W K, Wu Z G. Optimal decay rate of solutions for CahnHilliard equation with inertial term in multidimensions[J]. J Math Anal Appl, 2012, 387(1): 349358. [CrossRef] [MathSciNet] [Google Scholar]
 Li N Y, Mi L F. Pointwise estimate of solutions for the CahnHilliard equation with inertial term in multidimensions[J]. J Math Annal Appl, 2013, 397(1): 7587. [CrossRef] [Google Scholar]
 Wang W K, Yang T. The pointwise estimates of solutions for Euler equations with damping in multidimensions[J]. J Differential Equations, 2001, 173(2): 410450. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
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