On a viscous fourth-order parabolic equation with boundary degeneracy

A viscous fourth-order parabolic equation with boundary degeneracy is studied. By using the variational method, the existence of a time-discrete fourth-order elliptic equation with homogeneous boundary conditions is solved. Moreover, the existence and uniqueness for the corresponding parabolic problem with nondegenerate coefficient is shown by several asymptotic limit processes. Finally, by applying the regularization method, the existence and uniqueness for the problem with degenerate boundary coefficient is obtained by applying the energy method and a small parameter limit process.


Introduction
Many physical phenomena can be described by nonlinear fourth-order parabolic equations. The Cahn-Hilliard equation can be used to establish the model for phase transformation theory (see [3]). The degenerate fourth-order parabolic equations can show the motion of a very thin layer of a viscous compressible fluid (see [2,12], and [8]). Specially, in materials science, the epitaxial growth of nanoscale thin films can be given by nonlinear fourth-order parabolic equations (see [23] and [6]).
For the research of fourth-order parabolic equations, Liu [10] studied a Cahn-Hilliard equation with a zero-mass flux boundary condition, and the global existence of classical solutions with a nondegenerate m(w) and small initial energy was shown. Xu and Zhou [18] considered a nonlinear fourth-order parabolic equation with gradient degeneracy, and the corresponding existence of weak solutions was studied in the sense of distribution. For the nonlinear source problem, the existence and asymptotic behavior of solutions were given by Liang and Zheng in [9]. In the paper, we consider a viscous fourth-order parabolic equation with boundary degeneracy conditions. For the boundary degeneracy problem, there have been some research results about second-order equations. Yin and Wang (see [21] and [22]) gave the existence of weak solutions for a second-order singular diffusion problem, and the corresponding diffusion coefficients were allowed to degenerate on a portion of the boundary. For the boundary degeneracy problem with a gradient flow, Zhan in [24] obtained the existence and stability of solutions.
In the paper, a viscous fourth-order parabolic equation with boundary degeneracy is considered. If we drop the viscous term, the model can be treated as a thin film equation with a degenerate mobility rate. If the fourth-order diffusion term is replaced by a classic second-order diffusion, it often appeared in the research for pseudo-parabolic equations. For their research works, Xu and Su in [19] considered the initial-boundary value problem for a semi-linear pseudo-parabolic equation, and the corresponding global existence and finite time blow-up of solutions were given by potential well theory. In [7], a pseudoparabolic equation with a singular potential was shown. Moreover, the papers [20] and [16] studied the related nonlinear parabolic systems with power type source terms and time-fractional pseudo-parabolic problems. For the other references, the readers may refer to [4,11,13], and [14].
Our research problem with initial-boundary conditions has the following form: α > 0, p > 1, and γ > 0 are all constants. In physics, the capillarity-driven surface diffusion is from the term ( α (x)| w| p-2 w) (see Zangwill [23]). Here the function (x) is defined by = dist(x, ∂ ), which can yield the degeneration at ∂ . γ > 0 is the viscosity coefficient. We always suppose that the boundary ∂ is smooth enough and simple enough. Besides, for any constant σ ∈ (0, 1), the domain satisfies the condition -σ dx < ∞. The term γ w t denotes the viscous relaxation factor or viscosity. In order to obtain the existence of weak solutions for (1.1)-(1.3), we need to deal with the degenerate coefficient (x), and so we introduce the following approximate problem: where ε = +ε with ε > 0. From the existence of (1.4)-(1.6), we can conclude the existence of (1.1)-(1.3) by a limit process for ε → 0. The weak solution of (1.4)-(1.6) is shown in the following definition.

Definition 1 If a function w ε satisfies the conditions
then it is called a weak solution of (1.4)-(1.6).
Its existence is shown in the following proposition.

Definition 2 If a function w satisfies the conditions
The main existence is as follows.

) has a unique weak solution.
In the paper, C, C j (j = 1, 2, . . .) represent general constants, and the values may change from line to line. The paper is organized as follows. Section 2 gives the existence, uniqueness, and iterative estimates for the semi-discrete elliptic problem. In Sect. 3, we show the existence and uniqueness for the nondegenerate parabolic problem. The final section establishes the existence and uniqueness for the degenerate problem.

Elliptic problem
In this section, we introduce a semi-discrete problem, and some important iterative estimates are established. For the time interval [0, T], we make it into n subintervals with the equal width h = T n . Let w i = w(x, ih) and w 0 = w εI for i = 1, 2, . . . , n. We get the semi-discrete elliptic problem We will use the variational method to study the existence of (2.1)-(2.2), and so we define the functional as follows: for w i ∈ W 2,p 0 ( ). The corresponding existence result is shown in the following lemma.
Proof Young's inequality can give as l → +∞. Using Young's inequality again, we have It implies the estimate w kl W 2,p 0 ( ) ≤ C, and then we can seek a subsequence from {w kl } and a function w i ∈ W The weak lower semi-continuity yields . A standard procedure can show the existence of (2.1)-(2.2) (see [17] or [5]).
For the uniqueness, we suppose that w i1 and w i2 are two weak solutions, and we choose w i1w i2 as the test function to get Notice that, for arbitrary numbers ζ and η, the inequality |ζ | p-2 ζ -|η| p-2 η (ζη) ≥ 0 (2.12) holds if p > 1. Thus, one has w i1 = w i2 a.e. in .
To give the proof for the iterative estimates, we take w i as the test function and apply Young's inequality to find Thus, (2.5) and (2.6) have been shown. Meanwhile, taking w iw i-1 as the test function, we have Apply Young's inequality to give Therefore, a simple calculation can show assertions (2.7)-(2.9).

Parabolic problem with nondegenerate coefficient
In this section, we would give the proof of Proposition 1 for fixed constant ε > 0. We assume that w εI → w I in H 1 -norm as ε → 0. For convenience, we use the notation w to represent the weak solutions of (1.4)-(1.6).
For the purpose of existence, we define the following approximate solution: 0, elsewhere with i = 1, . . . , n.
For U (n) , we can establish the uniform estimates as follows.

Now we introduce another approximate solution
For V (n) , we establish the estimates as follows.
Next we give the proof of Proposition 1.
Proof of Proposition 1 Lemma 2 can ensure the existence of a subsequence of U (n) (we always take the same notation) and two functions w ∈ L ∞ (0, T; W 2,p 0 ( )) and v ∈ L p p-1 (Q T ) such that as n → ∞. Besides, from Lemma 3, we can find a subsequence of V (n) and a function such that On the other hand, for any ϕ ∈ C ∞ 0 (Q T ), we have as n → ∞ (i.e. h → 0). It implies w = a.e. in Q T and U (n) → w strongly in L 2 0, T; H 1 ( ) , If we perform the limit n → ∞ in the expression then we have for any ϕ ∈ C ∞ 0 (Q T ).
The next job is to prove v = | w| p-2 w. For each test function ψ ∈ C ∞ 0 (0, T), we define ϕ = ψw as the multiplier in (3.6) to get In (2.4), we use ψ(t)w i as the test function to give That becomes By introducing the notationŨ (n) ( For any function ϕ 1 ∈ C ∞ 0 (Q T ), we can seek two constants t 1 and t 2 with 0 < t 1 < t 2 < T such that supp ϕ 1 , supp ϕ 1 ⊂ (t 1 , t 2 ) × . Meanwhile, we redefine ψ as ψ ≡ 1 on (t 1 , t 2 ) and Similarly, one has Therefore, we have If we choose ζ = U (n) and η = (wλϕ 1 )) with λ > 0 and ϕ 1 ∈ C ∞ 0 (Q T ) in (2.12), then we have For other estimates in Proposition 1, we may apply J. Simon's lemma (see [15]) and Sobolev's embedding theorem (see [1] and [5]), and so we omit the details. The uniqueness can be shown as the corresponding proof of Lemma 1.

Existence for degenerate coefficient
For the solutions obtained in Proposition 1, we would use the notation w ε . In this section, we want to gain necessary uniform estimations with respect to ε so that the limit ε → 0 can be passed well.
For uniform estimates, we have the lemma.