Time-periodic solution for a fourth-order parabolic equation describing crystal surface growth

In this paper, by using the Galerkin method, the existence and uniqueness of time-periodic generalized solutions to a fourth-order parabolic equation describing crystal surface growth are proved.


Introduction
In the study of crystal surface growth, there arises the following diffusion equation u t = −j x + f (x, t), where u(x, t) denotes the variation of height from the average, j is the atom current parallel to the surface, and f (x, t) is a noise term caused by shot noise in the incoming flux.Taking j = u xxx + ux 1+|ux| 2 , we obtain the well-known BCF model (see [4,5,7,8,11,13]) = f (x, t), in (0, 1) × R. (1.1) During the past years, many authors have paid much attention to the equation (1.1).It was Rost and Krug [13] who studied the unstable epitaxy on singular surfaces using equation (1.1) with a prescribed slopedependent surface current.
In their paper, they derived scaling relations for the late stage of growth, where power law coarsening of the mound morphology is observed.In [11], in the limit of weak desorption, O. Pierre-Louis et al. derived the equation (1.1) for a vicinal surface growing in the step flow mode.This limit turned out to be singular, and nonlinearities of arbitrary order need to be taken into account.
Recently, H. Fujimura and A. Yagi [4] considered the equation of (1.1).In their paper, the uniqueness local solutions and the global solutions are obtained, a dynamical system determined from the initial-boundary value problem of the model equation was also constructed.In [5], H. Fujimura and A. Yagi continued a study on the model equation (1.1).They considered the asymptotic behavior of trajectories of the dynamical system by constructing exponential attractors and a Lyapunov function.There is much literature concerned with the Eq.(1.1), for more results we refer the reader to [6,14] and the references therein.
Furthermore, several authors have paid attention to the time-periodic problems [1,19,20].But, to the best of our knowledge, only a few papers deal with time periodic solutions of fourth-order diffusion equations.In [10,17], the existence of time periodic solutions for the Cahn-Hilliard type equation and viscous Cahn-Hilliard equation with periodic concentration dependent potentials and sources has been investigated.In [15,16], Wang et.al. considered 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 case.In [3], by using the Galerkin method and the Leray-Schauder fixed point theorem, Fu and Guo studied the existence and uniqueness of a time periodic solution for the viscous Camassa-Holm equation.There are also many papers were denoted to the periodic problems, for example [9,12,18] and so on.
Here, we investigate the existence and uniqueness of time-periodic generalized solutions to the equation (1.1) in one spatial dimension together with the condition and the time-periodic condition where ω > 0 is a constant and f (x, t) is ω-periodic functions with respect to the time t, which also satisfies Throughout this paper, we use the following notations.Let X be a Banach space, C k ω (R; X) denotes the set of X-valued ω-periodic functions on R with continuous derivatives up to order k.The norm in C k ω (R; X) is defined as where Let W k,p ω (R; X) denote the set of functions which belong to L p ω (R; X) together with their partial derivatives with respect to t up to the order k.
In the following, we frequently use the Poincaré inequality (see [2]): is the Galerkin approximate solution to the problem (1.1)- (1.3), where a group of function u Performing the Galerkin procedure for the equation (1.1), we obtain (2.1) with and the time-periodic condition Lemma 2.1 Suppose that f ∈ C ω (R; L 2 (0, 1)), M 1 = sup 0≤t≤ω f (•, t) .Then, there exists a approximate solution u N for problem (2.1)-(2.3),which satisfies where c 0 is a positive constant independent of N , M 1 .
Proof.Using Poincaré's inequality and Hölder's inequality, we have that is (2.4) Multiplying both sides of (2.1) by u N , and integrating it over (0, 1), making use of Hölder's inequality and (2.4), we obtain 1 2 Then, (2.5) It then follows from (2.6) that there is a t 1 ∈ (0, ω) such that (2.7) Adding (2.4) and (2.7) together gives (2.8) (2.9) Setting c 0 = 16 7 + 8ω, we complete the proof.Employing the Leray-Schauder fixed-point argument, we can prove that there exists at least one solution u Lemma 2.2 Suppose that the assumptions of Lemma 2.1 hold and where Here and in the sequel, Proof.Multiplying both sides of (2.1) by −u N xx , and integrating it over (0, 1), we obtain Then, using Hölder's inequality, we get EJQTDE, 2013 No. 7, p. 4 By Nirenberg's inequality, we have and Summing up, we get where (2.11) It then follows from (2.11) that there exists a time t 2 ∈ (0, ω) such that Then Integrating (2.10) again over [t 2 , t + ω] (∀t ∈ [0, ω]), using (2.13), we obtain Based on Sobolev's embedding theorem, we have Multiplying both sides of (2.1) by u N xxxx , and integrating it over (0, 1), we obtain Then, using Hölder's inequality, noticing that s ≤ 1 2 (1 + s 2 ), we get Using Nirenberg's inequality, we derive that Summing up, we deduce that where (2.17) It then follows from (2.17) that there exists a time t 3 ∈ (0, ω) such that Based on Sobolev's embedding theorem, we have Multiplying both sides of (2.1) by u N xxxxxx , and integrating it over (0, 1), we obtain By Nirenberg's inequality, we have Summing up, we deduce that
Then, Theorem 3.1 is proved.