Spatially Inhomogeneous Solutions to the Nonlocal Erosion Equation with Two Spatial Variables
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Journal of Mathematical Sciences, Vol. 250, No. 1, October, 2020
SPATIALLY INHOMOGENEOUS SOLUTIONS TO THE NONLOCAL EROSION EQUATION WITH TWO SPATIAL VARIABLES D. A. Kulikov P. G. Demidov Yaroslavl’ State University 14, Sovetskaya St., Yaroslavl’ 150003, Russia kulikov d [email protected]
UDC 517.9
We consider the periodic boundary value problem for the nonlocal erosion equation with two spatial variables and obtain sufficient conditions for the existence and stability of spatially inhomogeneous cycles. We analyze the boundary value problem in the case where the length of the domain is essentially greater than the width and obtain conditions for the existence of sufficiently many spatially inhomogeneous cycles depending on both spatial variables. For narrow domains the problem is reduced to analyzing an auxiliary boundary value problem for the Ginzburg–Landau equation. Bibliography: 11 titles.
1
Introduction
We study the mathematical model of topography formation on the semiconducter surface caused by ion bombardment. It is assumed that the ion flow is homogeneous and its directional line is perpendicular to the Oy-axis and forms an angle α with the Ox-axis. Normalizing the model, in particular, passing to dimensionless variables, we can write the model as [1, 2] uξ + b2 ( uξ )2 + b3 ( u ξ )3 , uτ = dΔu + b
(1.1)
where u = u(τ, ξ, η), Δu = uξξ + uηη , d is the diffusion coefficient, b, b2 , b3 ∈ R characterizing the ion flow intensity depend on α, and d, b > 0. Finally, u = u(τ, ξ + hξ , η) and hξ > 0. Regarding the sample size, we assume that 2lξ , 2lη > 0 and the conditions on the opposite sides of the rectangular are identical. Hence Equation (1.1) is considered together with the periodic boundary conditions (1.2) u(τ, ξ, η) = u(τ, ξ + 2lξ , η) = u(τ, ξ, η + 2lη ). From the technical point of view it is convenient to set π x=ξ , lξ
y=η
π , lη
t=τ
π2 . lξ2
Translated from Problemy Matematicheskogo Analiza 104, 2020, pp. 39-45. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2501-0042
42
Then we have the boundary value problem ux + a2 ( ux )2 + a3 ( u x )3 , ut = dΔp u + c
(1.3)
u(t, x + 2π, y) = u(t, x, y + 2π) = u(t, x, y),
(1.4)
= where Δp u = uxx + μuyy , μ = (lξ /lη )2 > 0, c = b(lξ /π) > 0, a2 = b2 , a3 = b3 (π/lξ ), u u(t, x + h, y), h = hξ (π/lξ ) > 0. Instead of the problem (1.3), (1.4), it is convenient to deal with an auxiliary problem for v = ux . Differentiating (1.3) with respect to x, we obtain the following problem for v = v(t, x, y):
where
vt = dΔp v + cwx + a2 (w2 )x + a3 (w3 )x ,
(1.5)
v(t, x + 2π, y) = v(t, x, y + 2π) = v(t, x, y), M0 (v) = 0,
(1.6)
1 2 2π 2π v(t, x, y)dxdy, M0 (v) = 2π 0
w = v(t, x + h, y).
0
The fact that the spatial variables are unequal is usually caused by the peculiarities of the ion flow. We note that the problem (1.5), (1.6) admits solutions v(t, x, y) = v1 (t, x) depending only on t and x, as well as solutions v2 (t, y) depending only on t and y. Then v2 (t, y) satisfies the equation v2t = dμv2yy , where v2 = v2 (t, y), v2 (t, y + 2π)
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