Pointwise Estimate of Cahn-Hilliard Equation with Inertial Term in R 3

: Cauchy problem of Cahn-Hilliard equation with iner‐ tial term in three-dimensional space is considered. Using delicate analysis of its Green function and its convolution with nonlinear term, pointwise decay rate is obtained.


Introduction
This paper is devoted to pointwise estimate of following Cahn-Hilliard equation with inertial term: ì í î Here D is the usual Laplace operator, η > 0 is a given constant.The nonlinear term f (u) has the form |u| θ + 1 or |u| θ -q + 1 × u q , where θ, q are positive integers and θq + 1 ≥ 0 θ ≥ 1.
Equation (1) is closely related to the well-known Cahn-Hilliard system System ( 2) is a hyperbolic equation with relaxation which describes phase separation of a binary mixture and u 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 ηu tt to (2).The unknown u 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 η = 1.
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][13][14] .Wang and Wu [15] took advantage of frequency decomposition and energy method, and they got global existence and L 2 decay rate of classical solution of (1) for the case of n ≥ 3 with small initial data.Based on their work, Li and Mi [16] got pointwise decay estimate of the solution for n ≥ 4. Their decay rates are closely related to the space dimension n.The solution decays faster if n is larger which makes it much more difficult to deal with lower space dimension.Obviously, compared with n ≥ 4 n = 3 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 C or C(x) a constant or constant depending on variable x.L p  W mp denote usual Lebesgue and Sobolev spaces on R n and , where i is the imaginary unit.Thus the inverse Fourier transform to the variable ξ of f ̂(ξt) is defined as 2 ∫f ̂(ξt)e ix × ξ dξ.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.

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 for some small ε, l ≥ 6, the Cauchy problem (1) admits a unique, global, classical solution u(xt) satisfying: The Green function of ( 1) is defined as 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 G(xt).Set For G 1 (xt)G 2 (xt), 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 C(N), such that 1e (λ --λ + )t λ +λ - × λ + e λ + t + e λ -t  using the same method of Theorem 2, we can get the same conclusion, that is Theorem 3 [16] There exists positive constant C(N), such that When |ξ| is large enough, using Taylor expansion, we get In order to get estimate of high frequency part, we need to understand the construction of G 3 (xt), and we can use Lemma 2.5 in Ref. [17].That is Lemma 1 [17] Assume that suppf ̂Ì where δ(x) is the Dirac function.Furthermore, for a positive integer Ì{x; |x| < 2ε 0 } with ε 0 being sufficiently small.From ( 4), ( 5) and Lemma 1, we have the following construction where f 1 (x) f 2 (x) satisfy Lemma 1.

Decay Estimation
We use (3) and decay rate of G to estimate the solution u(xt) step by step.Theorem 4 If u 0 (x), u 1 (x) satisfy the condition of Theorem 1, and Proof Here and afterwards we take N > 2r > 3 2 .We divide the following integral into three parts. If , we have 4 ) 4 Thus When |x| 4 ≤ t, we have B r (xt)≥ 1 2 , then From ( 7), ( 9), (10) and Theorem 2, we have Suppose function v Î H l  |α| ≤ l, from (4), ( 5), we have Take β = 0, we have From ( 13), ( 14), we have We know From ( 11), ( 16), the theorem is proved.Using the same method as that of Theorem 4, we also get the following theorem.