On the solutions of a fourth order parabolic equation modeling epitaxial thin film growth

Abstract

In this paper, we study the existence and uniqueness of global weak solution, the regularity of the solutions and the existence of global attractor for a fourth order parabolic equation modeling epitaxial thin film growth with Neumann boundary conditions in two space dimensions.

On the well-posedness of a nonlinear fourth-order extension of Richards' equation

“…In the same way, if p ∈ H 1 (Ω × (0, T )) is a weak solution of the fourth-order problem (6) and (7) as in Definition 15, then the Kirchhoff-transformed u = ψ(p) ∈ L 2 (0, T ; H 2 0 (Ω)) is a weak solution of the transformed fourth-order problem (9) and (10). This implies that the fourth-order problem (6), (7) and the transformed fourth-order problem (9) and (10) (6) and (7).…”

“…For γ = 0, the wellposedness of of ( 9) is proved in [1,12]. The wellposedness of other fourth-order parabolic equations describing thin film growth is investigated in [2,7,11,14]. Due to the nonlinearity of the first term on the left side of equation ( 9), we follow [1] and define the Legendre transform B for the primitive of b,…”

“…In this section, we utilize the well-posedness of the transformed problem ( 9) and (10) to prove the well-posedness of the fourth-order model ( 6) and (7). For this, we stress that the coefficients S, S ′ , K f are strictly positive.…”

“…Theorem 16. Assume that the initial condition in (7) satisfies p 0 ∈ H 2 0 (Ω) and the saturation function S ∈ C 1 (R) is Lipschitz continuity, strictly positive, and strictly monotone increasing. Assume also that the conductivity function K f ∈ C 1 (R) is strictly positive, bounded, and monotone increasing.…”

“…Let u ∈ H 1 (Ω × (0, T )) ∩ L 2 (0, T ; H 2 0 (Ω)) be the weak solution of the transformed problem (9) and (10). Then p = ψ −1 (u), where ψ is Kirchhoff's transformation, is the unique weak solution of the fourth-order problem (6) and (7) according to Definition 15. Proof. Using equation (8), Lemma 12, the boundedness and the strict positivity of K f , we have…”