Об асимптотической классификации решений не линейных уравнений третьего и четвертого порядков со степенной нелинейностью

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Theorem 1. For any real k > 1 and p 0 > 0 there exist positive
constants C1 and C2 such that local extrema of any typical maximally
extended to the right solution y(x) to equation (1) in some neighborhood of
the right bound x∗ of its domain satisfy the inequalities C1( x∗ −x0
j) − 4
k−1 ≤
≤ y( x0
j ) ≤ C2( x∗ − x0
j )− 4
k−1 .
J Let x0
J and x0
J +1 be two neighboring points of local extremum of
a solution y(x) such that the statements of Lemmas 8, 10, and 11 hold.
According to these Lemmas, for all j ≥ J we have
x0
j+1 − x0
j ≤ MFy (x0
j) −(k−1) ≤ MFy(x0
J )−( k−1)q−3(k−1)(j−J) ,
which implies x∗ − x0
J =
∞ Xj=J
x0
j+1 − x0
j ≤
MFy (x0
J ) −( k−1)
1 − q−3(k−1)
and
|y(x0
J )| (x∗ − x0
J )
4
k −1 ≤
Fy(x 0
J)4
ρ
MFy(x0
J )−(k−1)
1 − q−3(k−1)
4
k−1
=
=  Mp−
1
4
0
1 − q −3(k −1)  4
k−1
.
On the other hand, x0
j +1 −x0
j ≥ mF y(x 0
J )− (k−1)Q−3(k−1)(j−J ), which implies
|y(x0
J )| (x∗ − x0
J )
4
k −1 ≥ θFy(x0
J )4 mF y(x0
J )−(k−1)
1 − Q−3(k−1)
4
k−1
=
= θ
m
1 − Q−3(k −1)
4
k−1
. I
Asymptotic Classification of the Solutions to the Fourth-Order
Equation (1). In this part we consider the asymptotic behavior of nontrivial
solutions to equation (1) in the cases not previously considered. Then
asymptotic classification of all maximally extended solutions to equation
(1) will be given.
First for solutions to equation (1) generating in R4 curves lying entirely
in Ω+ , we describe their asymptotic behavior near the left boundary of the
domain.
Lemma 12. Suppose y( x) is a maximally extended to the left nontrivial
solution to equation (1) with derivatives changing their signs according to
scheme (7). Then the domain of y (x ) is unbounded to the left, the functions
y(x), y 0( x), y 00 (x), y 000(x) tend to zero as x → −∞, and the distance
between its neighboring zeros tends monotonically to ∞ as x → −∞.
18 ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2015. № 2