Abstract
The Kuramto–Sivashinsky equation with anisotropy effects models the spinodal decomposition of phase separating systems in an external field, the spatiotemporal evolution of the morphology of steps on crystal surfaces and the growth of thermodynamically unstable crystal surfaces with strongly anisotropic surface tension. Written in terms of the step slope, it can be represented in a form similar to a convective Cahn–Hilliard equation. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation.
Highlights
We investigate the well-posedness of the following Cauchy problem: ⎧ ⎪⎨∂tu + α∂x2u + β2∂x4u − γ2(∂xu)2∂x2u + τ ∂xu∂x2u
The function u is the concentration of one of the components of an alloy. [51] shows that (1.6) has an exact solution that describes the final stage of the spinodal decomposition, the formation of the interface between two stable state of an alloy with different concentrations
The main result of this paper is the following theorem
Summary
We investigate the well-posedness of the following Cauchy problem: ⎧ ⎪⎨∂tu + α∂x2u + β2∂x4u − γ2(∂xu)2∂x2u + τ ∂xu∂x2u. ⎪⎩u(0+,κx()∂=xuu)40(+x)q,(∂xu)2 + δ∂xu∂x3u = 0, t > 0, x ∈ R, x ∈ R, with α, β, γ, τ, κ, q, δ ∈ R, β, γ = 0, such that δ2 < 4β2γ2. We assume u0 ∈ H (R), Observe that, using the variable (see [24,57]). Equation (1.1) is equivalent to the following one:. Which is known as the convective Cahn–Hilliard equation (see [24,33]). From a physical point of view, (1.1) and (1.5) model the spinodal decomposition of phase separating systems in an external field [19,42,64], the spatiotemporal evolution of the morphology of steps on.
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