Abstract
This paper is concerned with a sixth-order diffusion equation, which describes continuum evolution of film-free surface. By using the regularity estimates for the semigroups, iteration technique and the classical existence theorem of global attractors we verified the existence of global attractor for this surface diffusion equation in the spaces H3(Ω) and fractional-order spaces Hk(Ω), where 0 ≤ k < ∞.
Highlights
In order to describe the continuum evolution of the film-free surface, the authors in [5] proposed the following classical surface diffusion equation: υn = D∆Sμ = D∆S(μγ + μω) = D∆S γαβCαβ + ν∆2u + μω, where υn, Ds, S0, Ω0, V0, R and T are the normal surface velocity, the surface diffusivity, the number of atoms per unit area on the surface, the atomic volume, the molar volume of lattice cites in the film, the universal gas constant and the absolute temperature, respectively
In order to consider the global attractors for Eq (1) in the Hk space, we introduce the definition as follows: H = u ∈ L2(Ω): u dx = 0, (16)
In order to study the long-time behavior of solutions, we prove the existence of global attractor of Eq (1) with boundary and initial value conditions
Summary
In [5], the authors show that wetting interactions between the film and the substrate can suppress this instability and qualitatively change its spectrum, leading to the damping of long-wave perturbations and the selection of the preferred wavelength at the instability threshold This creates a possibility for the formation of stable regular arrays of quantum dots even in the absence of epitaxial stresses. By using the regularity estimates for the semigroups, iteration technique and the classical existence theorem of global attractors we consider the global attractor of solutions for the initial boundary value problem for Eq (1). We give the following lemma on global existence and uniqueness of solution to problem (1)–(3). Combining Lemma 2 with Lemma 3, by [14] we have completed the proof of Theorem 1
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