EXISTENCE OF PERIODIC SOLUTION FOR A CAHN-HILLIARD/ALLEN-CAHN EQUATION IN TWO SPACE DIMENSIONS

. In this paper, we discuss the existence of the periodic solutions of a Cahn-Hillard/Allen-Cahn equation which is introduced as a simpliﬁcation of multiple microscopic mechanisms model in cluster interface evolution. Based on the Schauder type a priori estimates, which here will be obtained by means of a modiﬁed Campanato space, we prove the existence of time-periodic solutions in two space dimensions. The uniqueness of solutions is also discussed.

The equation (1.1) is introduced as a simplification of multiple microscopic mechanisms model [4] in cluster interface evolution. Karali and Ricciardi in [3] constructed special sequences of solutions to the equation (1.1) with f (x, t) = 0, converging to the second order Allen-Cahn equation. They considered the evolution equation without boundary, as well as the stationary case on domains with Dirichlet boundary conditions. Karali and Katsoulakis [4] discussed microscopic models describing pattern formation mechanisms for a prototypical model of surface processes that involve multiple microscopic mechanisms. Karali and Nagase [5] proved that the initial-boundary-value problem for (1.1) with f (x, t) = 0 admits a global smooth solution.
The time periodic solutions are important for the higher-order parabolic equation. During the past years, many authors have paid much attention to the time periodic solutions of higher order parabolic equations. Liu and Wang [6] proved the existence of time-periodic solutions for a sixth order nonlinear parabolic equation in two space dimensions. Yin et al. [9] considered the existence of time periodic solutions for the Cahn-Hilliard type equation in one dimension, see also [10]. Using the Galerkin method and the Leray-Schauder fixed point theorem, Wang et.al. [8] proved the existence and uniqueness of time-periodic generalized solutions and time-periodic classical solutions to the generalized Ginzburg-Landau model equation in 1D and 2D cases, see also [1]. However, many physical phenomena, such as the diffusion of oil film over a solid surface, need to be discussed in two dimensions and higher dimensions. Therefore, we should study the multi-dimensional case considered not only from mathematics itself but also from physical background. As far as we know, there are few investigations concerned with the time periodic solutions of such kind of equations.
This paper is a step further in the study of the [9]. The purpose is to prove the existence of time periodic solutions of the problem in two space dimensions. The main difficulties for treating the problem (1.1)-(1.3) are caused by the nonlinearity of both ∆ϕ(u) and ϕ(u). The main method that we use is based on the Schauder type a priori estimates, which here will be obtained by means of a modified Campanato space. After proving the compactness of the operator and some necessary estimates of the solutions, we obtain a fixed point of the operator in a suitable functional space, which is the desired solution of the problem (1.1)-(1.3). Compared the methods of this paper with [6], the u σ ∞ can be easily obtained in [6]. But in this paper, firstly, we have to get u σ q ≤ C(q < ∞) and then use the Gagliardo-Nirenberg inequality. On the other hand, to prove the main Theorem, the key step is to get a priori estimates on the Hölder norm of u in [6]. For this purpose, we need to consider the linear problem in [6] ∂u In this paper, we want to establish the Hölder norm of the ∇u. For this purpose, we consider the other linear problem Our main purpose is to find the relation between the Hölder norm of the solution ∇u and a(x, t), F (x, t). The method used in [6] is not applicable to the present situation. Due to the equation is fourth order equation, however, we want to establish the higher order Hölder norm of the solution than [6], and there are more difficulties. Based on a suitable integral inequality and Campanato spaces, we obtain the Hölder norm of the ∇u.

EXISTENCE OF PERIODIC SOLUTION FOR A CAHN-HILLIARD/ALLEN-CAHN 221
The plan of the paper is as follows. We first present a key step for the a priori estimates on the Hölder norm of solutions in Section 2, and then give the proof of our main theorem subsequently in Section 3.
Throughout the paper, the norms of L ∞ (Ω), L 2 (Ω), L s (Ω) are denoted by · ∞ , · and · s . We will only prove the existence of weak solutions in the space C 2+α, α 4 (Q T ), since the regularity follows a quite standard method [7]. To prove the existence of such solutions we employ the Leray-Schauder fixed point theorem which enables us to study the problem by considering the following equation where σ is a parameter taking values in the interval [0, 1], and g(x, t) ∈ C 2+α, α 4 (Q T ) in time t with period T . For any given function g(x, t) ∈ C 2+α, α 4 (Q T ), from the classical theory, we see that the problem admits a unique solution u σ ∈ C 4+α,1+ α 4 (Q T ) ⊂ C 2+α, α 4 (Q T ). Hence, we can define a map G as follows Obviously, for any given g ∈ C 2+α, α 4 (Q T ), G(g, 0) = 0. By virtue of the Leray-Schauder fixed point theorem, to prove the existence of solutions of the problem (1.1)-(1.3), we need to show that the map G is compact and prove that if u σ = G(g, σ) admits a fixed point u σ in the space C 2+α, α 4 (Q T ) for some σ ∈ [0, 1], then u σ C 2+α, α 4 (Q T ) ≤ C with C being a constant independent of u σ and σ.
where C is a constant independent of the solution and σ.
Proof. Multiplying (2.2) by u σ , integrating the result over Q T and using the conditions (1.2) and (1.3), we have

CHANGCHUN LIU AND HUI TANG
By B(t) ≤ K, we see that We notice that From the above inequality and the assumptions on The above inequality implies that On the other hand, by the assumptions M ≤ A(t) ≤ M , B(t) ≤ K and (2.5), we have Therefore, we get By the Young inequality, we obtain Combining the above estimate with (2.4) and (2.5), we have On the other hand, integrating by parts and noticing (2.2) and (1.2), we obtain Integrating the above inequality over (0, T ) and noticing the periodicity of H, we have

CHANGCHUN LIU AND HUI TANG
Integrating | dH dt | over (0, T ), using(2.7), (2.8) and (2.9), we have By virtue of (2.10) and (2.11), we know that Combining the above estimate with the boundary value conditions, we obtain that By the Gagliardo-Nirenberg inequality (noticing that we consider only the twodimensional case), we can obtain ∇u σ 4 ≤ C ∆u σ 2 .

EXISTENCE OF PERIODIC SOLUTION FOR A CAHN-HILLIARD/ALLEN-CAHN 225
Therefore, by (2.9), we have (2.14) Similarly, multiplying (2.2) by ∆ 2 u σ , integrating the result over Q T and using the conditions (1.2) and (1.3), we have By Hölder's inequality, we know By the Gagliardo-Nirenberg inequality, we obtain Therefore, we have
Here, we have used the mean value theorem, where x * = y * + θ * (∆t) 1/8 . Hence by the Hölder inequality, (2.13), (2.14), (2.15), (2.17) and Hence u(x, 0) = u(x, T ). (3.3) Without loss of generality, we may assume that a(x, t) and F (x, t) are sufficiently smooth, otherwise we replace them by their approximation functions. Our main purpose is to find the relation between the Hölder norm of the solution ∇u and a(x, t), F (x, t).
Let u be the solution of the problem (3.1), (3.2), (3.3). We split u on S R into u = u 1 + u 2 , where u 1 is the solution of the problem ∂u 1 ∂t + γ∆ 2 u 1 + a(x 0 , t 0 )∆u 1 = 0, (x, t) ∈ S R , (3.4) (3.6) and u 2 solves the problem By classical linear theory, the above decomposition is uniquely determined by u.
We need several lemmas on u 1 and u 2 .
Proof. Multiply the equation (3.7) by ∇u 2 and integrate the resulting relation over (t 0 − R 4 , t) × B R (x 0 ). Integrating by parts, we have Noticing that we obtain the estimate and the proof is complete.
Proof. For simplicity, we only prove the first inequality, since the other inequality can be shown similarly. Choose a cut-off function Let g(t) ∈ C ∞ 0 (t 0 , +∞) with 0 ≤ g(t) ≤ 1, 0 ≤ g (t) ≤ C It follows from integrating by parts, By the Cauchy-Schwarz inequality, we have Similarly, we obtain Noticing that t t0−R 4 B R (x0) g(s)χ 2 |∇χ| 2 |∆u 1 | 2 dxds 232 CHANGCHUN LIU AND HUI TANG Combining the above expressions yields We obtain immediately the desired first inequality of the lemma and the proof is completed.
Then for any ρ ∈ (0, R), Proof. One only needs to check the inequality for ρ ≤ R 2 . From Lemma 3.2 and Lemma 3.3, we have 1 On the other hand, The conclusion of the lemma follows at once.
Proof. By Lemma 3.4, The conclusion follows immediately from the Proposition 1.3 in [2].