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High-Speed Transmission in Long-Haul Electrical Systems.

1. Introduction

We study the equations governing the high-speed transmission in long-haul electrical systems

[mathematical expression not reproducible] (1)

where [lambda] [member of] R, [absolute value of [[[partial derivative].sub.x].sup.3] = [F.sup.-1] [[absolute value of [xi]].sup.3] F, and F is the Fourier transformation defined by F[phi] = (1/[square root of 2[pi]]) [[integral].sub.R] [e.sup.-ix[xi]] [phi]dx. Note that we have the relation u(-t, x) = [bar.u](t, -x), so we can only consider the case t > 0. For the regular solution of (1) we have the conservation law [mathematical expression not reproducible]. We are interested in the case of nonzero mass condition [[integral].sub.R] [u.sub.0](x)dx = 0. By (1) we get the conservation of the mass [[integral].sub.R] u(t, x)dx = [[integral].sub.R] [u.sub,0](x)dx [not equal to] 0 for all t > 0.

This equation arises in the context of high-speed soliton transmission in long-haul optical communication system [1]. Also it can be considered as a particular form of the higher order nonlinear Schrodinger equation introduced by [2] to describe the nonlinear propagation of pulses through optical fibers. This equation also represents the propagation of pulses by taking higher dispersion effects into account than those given by the Schrodinger equation (see [3-11]).

The higher order nonlinear Schroodinger equations have been widely studied recently. For the local and global well-posedness of the Cauchy problem we refer to [12-14] and references cited therein. The dispersive blow-up was obtained in [15]. The existence and uniqueness of solutions to (1) were proved in [16-25] and the smoothing properties of solutions were studied in [18-21, 24, 26-31]. The blow-up effect for a special class of slowly decaying solutions of Cauchy problem (1) was found in [32].

As far as we know the question of the large time asymptotics for solutions to Cauchy problem (1) is an open problem. We develop here the factorization technique originated in our previous papers [33-38].

We denote the Lebesgue space by [mathematical expression not reproducible]. The weighted Sobolev space [mathematical expression not reproducible]. We also use the notations [H.sup.m,s] = [H.sup.m,s.sub.2], [H.sup.m] = [H.sup.m,0] shortly, if it does not cause any confusion. Let C(I; B) be the space of continuous functions from an interval I to a Banach space B. Different positive constants might be denoted by the same letter C. We denote by or F[phi] [??]([xi]) = (1/[square root of 2[pi]]) [[integral].sub.R] [e.sup.-ix[xi]] ([phi])dx the Fourier transform of the function [phi], then the inverse Fourier transformation is given by [F.sup.-1][phi] = (1/[square root of 2[pi]]) [[integral].sub.R] [e.sup.-ix[xi]] [phi]([xi])d[xi].

We are now in a position to state our result.

Theorem 1. Assume that the initial data [u.sub.0] [member of] [H.sup.1] [intersection] [H.sup.0,1] have a sufficiently small norm [mathematical expression not reproducible]. Then there exists a unique global solution [mathematical expression not reproducible] of Cauchy problem (1). Furthermore the estimate

[mathematical expression not reproducible] (2)

is true, where p > 4.

Next we prove the existence of the self-similar solutions [v.sub.m](t, x) = [t.sup.-1/3] [f.sub.m]([xt.sup.-1/3]).

Theorem 2. There exists a unique solution of Cauchy problem (1) in the self-similar form [v.sub.m](t, x) = [t.sup.-1/3] [f.sub.m]([xt.sup.-1/3]), such that

[mathematical expression not reproducible], (3)

where m is sufficiently small number and

[mathematical expression not reproducible]. (4)

Furthermore the estimate

[mathematical expression not reproducible]. (5)

is true, where p > 4.

Now we state the stability of solutions to Cauchy problem (1) in the neighborhood of the self-similar solution [v.sub.m] (t, x).

Theorem 3. Suppose that

[mathematical expression not reproducible]. (6)

Let u(t, x) and [v.sub.m](t, x) be the solutions constructed in Theorems 1 and 2, respectively. Then there exists small [gamma] > 0 such that the asymptotics

[mathematical expression not reproducible] (7)

are true for t [greater than or equal to] 1.

Our approach is based on the factorization techniques. Define the free evolution group [mathematical expression not reproducible] and write

[mathematical expression not reproducible], (8)

where [D.sub.t][phi] = [[absolute value of t].sup.-1/2] [phi](x/t) is the dilation operator. There is a unique stationary point [mathematical expression not reproducible], which is defined as the root of the equation [xi][absolute value of [xi]] = x for all x [member of] R. Define the scaling operator (B[phi])(x) = [phi]([mu](x)). Hence we find the following decomposition U(t)[F.sup.-1] [phi] = [D.sub.t]BM[V.sub.[phi]], where the multiplication factor [mathematical expression not reproducible] and the deformation operator

[mathematical expression not reproducible], (9)

where the phase function [mathematical expression not reproducible]. We have [A.sub.1] = [A.sub.0] + i[eta], and also [A.sub.1]V = Vi[xi], [in, V] = -[A.sub.0]V; therefore we obtain the commutator [[partial derivative].sub.[eta]]V = -2t[absolute value of [eta]][in, V]. Since [[partial derivative].sub.[xi]]([xi], [eta]) = [xi][absolute value of [xi]] - [eta][absolute value of [eta]] then we get it [[eta][absolute value of [eta]], V][phi] = -V[[partial derivative].sub.[xi]][phi]. Also we need the representation for the inverse evolution group FU(-t)[phi] = [V.sup.*][bar.M][B.sup.-1][D.sup.-1.sub.t], where the inverse dilation operator [D.sup.-1.sub.t][phi] = [[absolute value of t].sup.1/2] [phi](xt), the scaling operator ([B.sup.-1] [phi])([eta]) = [phi]([eta][absolute value of [eta]]), and the inverse deformation operator

[mathematical expression not reproducible]. (10)

We have i[xi][V.sup.*][phi] = [V.sup.*] Hence the commutator [i[xi], [V.sup.*]] = [V.sup.*] [A.sub.0]. Define the new dependent variable [??] = FU(-t)u(t). Since FU(-t)L = [[partial derivative].sub.t]FU(-t) with L = [[partial derivative].sub.t] + (i/3)[absolute value of [[[partial derivative].sub.t]].sup.3], applying the operator FU(-t) to (1), substituting [mathematical expression not reproducible], and using the factorization techniques, we get

[mathematical expression not reproducible] (11)

since the nonlinearity is gauge invariant. Finally we mention some important identities. The operator J = U(t)xU(-t) = x + it[[partial derivative].sub.t][absolute value of [[partial derivative].sub.t]] plays a crucial role in the large time asymptotic estimates. Note that J commutes with L, that is, [J, L] = 0. To avoid the derivative loss we also use the operator P = 3t[[partial derivative].sub.t] + [[partial derivative].sub.t]x. Note the commutator relation [mathematical expression not reproducible] with [mathematical expression not reproducible] holds.

2. Estimates in the Uniform Norm

2.1. Kernels. Define the kernel

[mathematical expression not reproducible] (12)

for [eta] [not equal to] 0, where the cutoff function [THETA](z) [member of] [C.sup.2](R) is such that [THETA](z) = 0 for z [less than or equal to] 1/3 or z [greater than or equal to] 3 and [THETA](z) = 1 for 2/3 [less than or equal to] z [less than or equal to] 3/2. We change [xi] = [eta]y, then we get [mathematical expression not reproducible]. To compute the asymptotics of the kernel [A.sub.j](t, [eta]) for large t we apply the stationary phase method (see [39], p. 110)

[mathematical expression not reproducible] (13)

for z [right arrow] +[infinity], where the stationary point y0 is defined by the equation g'([y,sub.0]) = 0. By virtue of formula (13) with g(y) = -G(y), f(y) = [THETA](y)[y.sup.j], and [y,sub.0] = 1, we get

[mathematical expression not reproducible] (14)

for t[[eta].sup.3] [right arrow] [infinity]. Also we have the estimate [mathematical expression not reproducible].

In the same manner changing [eta] = [xi]y, we get for the kernel

[mathematical expression not reproducible] (15)

for [mathematical expression not reproducible]. Then by virtue of formula (13) with g(y) = [??](y), f(y) = [THETA](y)[absolute value of y], and [y.sub.0] = 1, we obtain

[mathematical expression not reproducible] (16)

for t[[xi].sup.3] [right arrow] [infinity]. Also we have the estimate [absolute value of [A.sup.*](t, [xi])] [less than or equal to] [Ct.sup.1/2] [[xi].sup.2] [<t[[xi].sup.3]>.sup.1/2].

2.2. Asymptotics for the Operator V. In the next lemma we estimate the operator V in the uniform norm. Define the cutoff function [[chi].sub.1](z) [member of] [C.sup.2](R) such that [[chi].sub.1](z) = 0 for [absolute value of z] [greater than or equal to] 3 and [[chi].sub.1](z) = 1 for [absolute value of z] < 2 and [[chi].sub.2](z) = 1 - [[chi].sub.1](z). Consider two operators

[mathematical expression not reproducible], (17)

so that we have [mathematical expression not reproducible].

Lemma 4. The following estimates [mathematical expression not reproducible] are valid for all t [greater than or equal to] 1, n [not equal to] 0.

Proof. We write

[mathematical expression not reproducible] (18)

for [eta] [not equal to] 0. For the summand I1 we integrate by parts via identity

[mathematical expression not reproducible] (19)

[mathematical expression not reproducible]. (20)

We find the estimates

[mathematical expression not reproducible] (21)

in the domain 1/3 < [xi]/[eta] < 3. Therefore we obtain

[mathematical expression not reproducible]. (22)

By the Hardy inequality [mathematical expression not reproducible] and by the Cauchy-Schwarz inequality, changing [xi] = [eta]y we find

[mathematical expression not reproducible]. (23)

To estimate the integral [I.sub.2] we integrate by parts via the identity

[mathematical expression not reproducible]. (25)

We find the estimates [mathematical expression not reproducible] and

[mathematical expression not reproducible]. (26)

Then by the Hardy inequality we obtain

[mathematical expression not reproducible]. (27)

We have [mathematical expression not reproducible].

To estimate [V.sub.2][[xi].sup.j] we integrate by parts via identity (24)

[mathematical expression not reproducible]. (28)

We find the estimates [mathematical expression not reproducible]. Then by the Hardy inequality we obtain

[mathematical expression not reproducible]. (29)

We have [mathematical expression not reproducible]. Lemma 13 is proved.

2.3. Asymptotics for the Operator [V.sup.*]. We next consider the operator [mathematical expression not reproducible], then by the Riesz interpolation theorem (see [40], p. 52) we have

[mathematical expression not reproducible] (30)

for 2 [less than or equal to] p [less than or equal to] [infinity]. In the next lemma we find the asymptotics of [V.sup.*]. Denote [??] = [xi][t.sup.1/3]. Also define the norm [mathematical expression not reproducible].

Lemma 5. Let [mathematical expression not reproducible] is valid for all t [greater than or equal to] 1.

Proof. We write

[mathematical expression not reproducible] (31)

for [xi] [not equal to] 0. In the integral [I.sub.1] we use the identity

[mathematical expression not reproducible] (32)

with [mathematical expression not reproducible], and integrate by parts

[mathematical expression not reproducible]. (33)

Then apply the estimates [mathematical expression not reproducible] then we find the Hardy inequality

[mathematical expression not reproducible]. (34)

Hence [mathematical expression not reproducible]. Therefore changing [eta] = y[xi], we have

[mathematical expression not reproducible] (35)

if 1/4 + 2[beta] [less than or equal to] [alpha] [less than or equal to] 5/2. In the integral [I.sub.2] using the identity

[e.sup.itS([xi],[eta])] = [H.sub.4] [[partial derivative].sub.[eta]]([eta][e.sup.itS([xi],[eta])]) (36)

with [mathematical expression not reproducible], we integrate by parts

[mathematical expression not reproducible]. (37)

Then using the estimates [[partial derivative].sub.[eta]]S([xi], [eta]) = O([absolute value of [eta]]([xi] + [absolute value of [eta]])) in the domains [eta]/[xi] < 1/3 and [eta]/[xi] [greater than or equal to] 3, we get

[mathematical expression not reproducible]. (38)

Therefore by the Hardy inequality

[mathematical expression not reproducible]. (39)

We have

[mathematical expression not reproducible]. (40)

if 1/4 + 2[beta] [less than or equal to] [alpha] < 5/2 - 2[beta], 0 [less than or equal to] [beta] < 1/2. Therefore we get [mathematical expression not reproducible]. Lemma 5 is proved.

3. Commutators with V

First we estimate the Fourier type integral

[mathematical expression not reproducible] (41)

in the [L.sup.2]-norm. In the particular factorized case q(t, [xi], [eta]) = [q.sub.1]([xi])[q.sub.2]([eta]),with estimate [absolute value of [q.sub.2](u(x))] [less than or equal to] [[absolute value of [mu](x)].sup.1/2], we find

[mathematical expression not reproducible]. (42)

Next we obtain a more general result.

Lemma 6. Suppose that [mathematical expression not reproducible].

Proof. We write

[mathematical expression not reproducible], (43)

where the kernel [mathematical expression not reproducible]. Integrating two times by parts via the identity [mathematical expression not reproducible] we get

[mathematical expression not reproducible]. (44)

Since [mathematical expression not reproducible] we get

[mathematical expression not reproducible]. (45)

Then by the Cauchy-Schwarz inequality and Young inequality we obtain

[mathematical expression not reproducible]. (46)

if v [member of] (0, 1). Lemma 14 is proved.

Next we estimate W[phi](0).

Lemma 7. Suppose that [mathematical expression not reproducible]. Then the estimate [mathematical expression not reproducible] is true.

Proof. As in the proof of Lemma 13 we decompose

[mathematical expression not reproducible] (47)

for [eta] = 0. In the first summand I1 we integrate by parts via identity (19), to get

[mathematical expression not reproducible]. (48)

Using the estimate [mathematical expression not reproducible] then changing [xi] = [eta]y, we obtain

[mathematical expression not reproducible]. (49)

Thus we get [mathematical expression not reproducible]. To estimate the integral I2 we integrate by parts via identity (24), to get

[mathematical expression not reproducible]. (50)

We find the estimates [mathematical expression not reproducible], then we obtain

[mathematical expression not reproducible]. (51)

Thus as above we get [mathematical expression not reproducible]. To estimate [I.sub.3] we integrate by parts via identity (24)

[mathematical expression not reproducible]. (52)

We find the estimates [mathematical expression not reproducible] in the domain [absolute value of [xi]] [greater than or equal to] (3/2) [absolute value of [eta]], and then we obtain

[mathematical expression not reproducible]. (53)

if 0 < [gamma] < l. Thus we find

[mathematical expression not reproducible] (54)

for all t [greater than or equal to] l. Lemma 7 is proved.

In the next lemma we estimate the commutator [[eta], [V.sub.1]]. Define the norm [mathematical expression not reproducible].

Lemma 8. Let j [greater than or equal to] 0, v [member of] (0, 1). Then the estimate [mathematical expression not reproducible] is true for all t [greater than or equal to] 1.

Proof. For [eta] [not equal to] 0 we integrate by parts

[mathematical expression not reproducible], (55)

where [mathematical expression not reproducible], we have

[mathematical expression not reproducible] (56)

and similarly [mathematical expression not reproducible], and by Lemma 14

[mathematical expression not reproducible] (57)

and by the Hardy inequality and Lemma 14

[mathematical expression not reproducible]. (58)

Also we have

[mathematical expression not reproducible] (59)

for all [xi], [eta] [member of] R, in the domain [absolute value of [xi]] [less than or equal to] 3[absolute value of [eta]]. Hence we get [mathematical expression not reproducible]. Therefore applying Lemma 7 we obtain [mathematical expression not reproducible]. Lemma 8 is proved.

In the next lemma we estimate the operator [V.sub.2].

Lemma 9. Let j = 0, 1, 2, v [member of] (0, 1). Then the estimate [mathematical expression not reproducible] is true for all t [greater than or equal to] 1.

Proof. We integrate by parts

[mathematical expression not reproducible], (60)

where [mathematical expression not reproducible]. We find

[mathematical expression not reproducible], (61)

and [mathematical expression not reproducible]. Hence by Lemma 14 we find

[mathematical expression not reproducible], (62)

and by the Hardy inequality

[mathematical expression not reproducible]. (63)

Also we have [q.sub.3](t, [xi], [eta]) = O([([absolute value of [eta]] + [absolute value of [xi]]).sup.-1]) for all [xi] [eta], in the domain [absolute value of [xi]] [greater than or equal to] (3/2) [absolute value of [eta]], if j = 0, 1, 2. Hence [mathematical expression not reproducible], and by Lemma7

[mathematical expression not reproducible]. (64)

Lemma 9 is proved.

In the next lemma, we estimate the derivative [[partial derivative].sub.[eta]]V.

Lemma 10. Let j = 0, 1, v [member of] (0,1). Then the estimate [mathematical expression not reproducible] is true for all t [greater than or equal to] 1.

Proof. Since [A.sub.1]V = Vi[xi] with [A.sub.1] = [A.sub.0] + i[eta], [A.sub.0] = (1/2t[absolute value of [eta]])[[partial derivative].sub.[eta]], then we obtain the [[partial derivative].sub.[eta]]V commutator = 2t[absolute value of [eta]][in, V]. Also V = [V.sub.1] + [V.sub.2]. Hence

[mathematical expression not reproducible]. (65)

By Lemma 8 we find [mathematical expression not reproducible]. Hence the result of the lemma follows. Lemma 10 is proved.

4. A Priori Estimates

Local existence and uniqueness of solutions to Cauchy problem (1) were shown in [19, 20] when [u.sub.0] [member of] [H.sup.1]. By using the local existence result, we have the following.

Theorem 11. Assume that the initial data [u.sub.0] [member of] [H.sup.1] [intersection] [H.sup.0,1]. Then there exists a unique local solution u of Cauchy problem (1) suchthat U(-i)u [member of] C([0, T]; [H.sup.1] [intersection] [H.sup.0,1]).

We can take T > 1 if the data are small in [H.sup.1] [intersection] [H.sup.0,1] and we may assume that [mathematical expression not reproducible]. To get the desired results, we prove a priori estimates of solutions uniformly in time. Define the following norm

[mathematical expression not reproducible], (66)

where J = U(i)xU(-i), [??](i) = FU(-i)u(i), and p > 4.

First we obtain the large time asymptotic behavior of the nonlinearity FU(-t)[[partial derivative].sub.x]([[absolute value of u].sup.2]u).

Lemma 12. The asymptotics [mathematical expression not reproducible] is small.

Proof. In view of factorization formula (11) we find [mathematical expression not reproducible]. Then by Lemma 5 with [alpha] = 1/2 + v, [beta] = 2v, and v > 0 small, we get

[mathematical expression not reproducible] (67)

in the case of [absolute value of [xi]] < [t.sup.1/3] and

[mathematical expression not reproducible] (68)

in the case of [absolute value of [xi]] > [t.sup.-1/3]. Via identity [mathematical expression not reproducible], we consider the remainder terms

[mathematical expression not reproducible], (69)

where [[psi].sub.1] = Vi[xi][??]. By Lemma 10 with j = 0,1, we have

[mathematical expression not reproducible]. (70)

Using Lemma [mathematical expression not reproducible]. Lemma 12 is proved.

Next we estimate the solutions in the norm [X.sub.T].

Lemma 13. Assume that [mathematical expression not reproducible] holds. Then there exists [epsilon] > 0 such that the estimate [mathematical expression not reproducible] is true for all T > 1.

Proof. By continuity of the norm [mathematical expression not reproducible], with respect to T, arguing by the contradiction we can find the first time T >0 such that [mathematical expression not reproducible]. To prove the estimate for the norm [mathematical expression not reproducible] we use (11). Then in view of Lemma 12, we get

[mathematical expression not reproducible]. (71)

For the case of [absolute value of [xi]] < [t.sup.-1/3] we can integrate [mathematical expression not reproducible]. For the case of [absolute value of [xi]] [greater than or equal to] [t.sup.-1/3] multiplying by [??] and taking the real part of the result we obtain [mathematical expression not reproducible]. Integrating in time we obtain

[mathematical expression not reproducible]. (72)

Therefore [mathematical expression not reproducible]. Applying estimate of Lemma 4 we find [mathematical expression not reproducible], and

[mathematical expression not reproducible] (73)

if p > 4. Consider a priori estimates for [mathematical expression not reproducible]. Using the identity [mathematical expression not reproducible]. Applying the operator [[partial derivative].sup.-1.sub.x] P to (1), in view of the commutators [mathematical expression not reproducible]. Then by the energy method we obtain

[mathematical expression not reproducible], (74)

from which it follows that [mathematical expression not reproducible]. Lemma 13 is proved.

5. Proof of Theorem 1

By Lemma 13 we see that a priori estimate [mathematical expression not reproducible] is true for all T > 0. Therefore the global existence of solutions of Cauchy problem (1) satisfying the estimate [mathematical expression not reproducible] follows by a standard continuation argument by local existence Theorem 11.

6. Proof of Theorem 2

In this section we prove the existence of a unique selfsimilar solution [v.sub.m](t, x) [equivalent to] [t.sup.-1/3] [f.sub.m]([xt.sup.-1/3]) for (1), which is uniquely determined by the total mass condition m = (1/[square root of 2[pi]]) [[integral].sub.R] [v.sub.m](t, x)dx [not equal to] 0. Define the operators

[mathematical expression not reproducible]. (75)

Then for the self-similar solutions [mathematical expression not reproducible]. Note that F([[phi].sub.m]) is not in [L.sup.2]. Therefore we need the approximate equation. Define [THETA]([eta]) = l for [absolute value of [eta]] [less than or equal to] 1 and [THETA]([eta]) = 0 for [absolute value of [eta]] > 2, and denote [[THETA].sub.R]([eta]) = [THETA]([eta]/R). Also define the approximate equation

[mathematical expression not reproducible]. (76)

Let us show a priori estimate [mathematical expression not reproducible] uniformly in R. Applying Lemma 12 with i =l we get

[mathematical expression not reproducible]. (77)

Integrating with respect to [eta], we obtain [mathematical expression not reproducible]. Also multiplying by [bar.[[phi].sub.m,R]] and integrating with respect to q we get [mathematical expression not reproducible]. Thus we obtain [[parallel] [[phi].sub.m,R] [parallel].sub.Z] [less than or equal to] 2[absolute value of m] + C[[parallel] [[phi].sub.m,R] [parallel].sub.3.sub.Z] [less than or equal to] 3[absolute value of m] for some small m. Taking the limit R [right arrow] [infinity], we find that there exists a unique solution [mathematical expression not reproducible]. By the definition of [mathematical expression not reproducible]. In the same way as in the proof of (73) we have [L.sup.p] estimate of [v.sub.m] for p > 4.

7. Proof of Theorem 3

Define the norm [mathematical expression not reproducible].

Lemma 14. Suppose that [mathematical expression not reproducible] be a self-similar solution. Then the estimate [mathematical expression not reproducible] is true for all T > l.

Proof. By the continuity of the norm [mathematical expression not reproducible] with respect to T, arguing by the contradiction we can find for the first time T > 0 such that [mathematical expression not reproducible]. Applying estimate of Lemma 4 we find

[mathematical expression not reproducible]. (78)

Thus we need to estimate the norm [mathematical expression not reproducible]. For the difference y we get from (1) L[[partial derivative].sup.-1.sub.x] Py = [lambda](P + 2)([[absolute value of [u.sub.1]].sup.2] [u.sub.1] - [[absolute value of [u.sub.2]].sup.2] [u.sub.2]). Hence by the energy method

[mathematical expression not reproducible]. (79)

Next we get

[mathematical expression not reproducible]. (80)

In the same manner

[mathematical expression not reproducible]. (81)

Note that [mathematical expression not reproducible] for the case of self-similar solution [u.sub.2] = [t.sup.-1/3] f(x[t.sup.-1/3]). Hence

[mathematical expression not reproducible]. (82)

By (78) we have [mathematical expression not reproducible] we use the above estimates to get

[mathematical expression not reproducible]. (83)

In view of Lemma 4

[mathematical expression not reproducible]. (84)

Since [??](0) = 0, we get by the Hardy inequality [mathematical expression not reproducible] and by a direct calculation [mathematical expression not reproducible]. Hence [mathematical expression not reproducible]. Lemma 14 is proved.

Now we turn to the proof of asymptotic formula (7) for the solutions u of Cauchy problem (1). Let [v.sub.m](t, x) be the self-similar solution with the total mass condition m = (1/[square root of 2[pi]) [[integral].sub.R] [u.sub.0](x)dx = (1/[square root of 2[pi]) [[integral].sub.R] [v.sub.m] (t, x)dx = 0. Note that [mathematical expression not reproducible]. Thus asymptotics (7) follows. Theorem 3 is proved.

https://doi.org/10.1155/2018/8236942

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The work is partially supported by CONACYT 252053-F and PAPIIT Project IN100817.

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Beatriz Juarez-Campos, (1) Elena I. Kaikina (iD), (2) Pavel I. Naumkin (iD), (2) and Hector Francisco Ruiz-Paredes (1)

(1) Instituto Tecnologico de Morelia, Avenida Tecnologico No. 1500, Lomas de Santiaguito, 58120 Morelia, MICH, Mexico

(2) Centro de Ciencias Matematicas, UNAM Campus Morelia, AP 61-3 Xangari, 58089 Morelia, MICH, Mexico

Correspondence should be addressed to Elena I. Kaikina; ekaikina@matmor.unam.mx

Received 29 January 2018; Accepted 12 March 2018; Published 18 April 2018

Academic Editor: Sining Zheng
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Title Annotation:Research Article
Author:Juarez-Campos, Beatriz; Kaikina, Elena I.; Naumkin, Pavel I.; Ruiz-Paredes, Hector Francisco
Publication:International Journal of Differential Equations
Article Type:Report
Date:Jan 1, 2018
Words:5361
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