- Research Article
- Open access
- Published:
Asymptotics for Nonlinear Evolution Equation with Module-Fractional Derivative on a Half-Line
Boundary Value Problems volume 2011, Article number: 946143 (2011)
Abstract
We consider the initial-boundary value problem for a nonlinear partial differential equation with module-fractional derivative on a half-line. We study the local and global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.
1. Introduction
We study the local and global existence and asymptotic behavior for solutions to the initial-boundary value problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ1_HTML.gif)
where ,
, and
is the module-fractional derivative operator defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ2_HTML.gif)
where is the Laplace transform for
with respect to
and
is the Heaviside function:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ3_HTML.gif)
The Cauchy problem for a wide class of nonlinear nonlocal dissipative equations has been studied extensively. In particular, the general approach for the study of the large time asymptotics to the Cauchy problem for different nonlinear equations was investigated in the book [1] and the references therein.
The boundary value problems are more natural for applications and play an important role in the contemporary mathematical physics. However, their mathematical investigations are more complicated even in the case of the differential equations, with more reason to the case of nonlocal equations. We need to answer such basic question as how many boundary values should be given in the problem for its solvability and the uniqueness of the solution? Also it is interesting to study the influence of the boundary data on the qualitative properties of the solution. For examples and details see [2–12] and references therein.
The general theory of nonlinear nonlocal equations on a half-line was developed in book [13], where the pseudodifferential operator on a half-line was introduced by virtue of the inverse Laplace transformation. In this definition it was important that the symbol
must be analytic in the complex right half-plane. We emphasize that the pseudodifferential operator
in (1.1) has a nonanalytic nonhomogeneous symbol
and the general theory from book [13] cannot be applied to the problem (1.1) directly. As far as we know there are few results on the initial-boundary value problems with pseudodifferential equations having a nonanalytic symbol. The case of rational symbol
which has some poles in the complex right half-plane was studied in [14, 15], where it was proposed a new method for constructing the Green operator based on the introduction of some necessary condition at the singular points of the symbol
. In [16] there was considered the initial-boundary value problem for a pseudodifferential equation with a nonanalytic homogeneous symbol
, where the theory of sectionally analytic functions was implemented for proving that the initial-boundary value problem is well posed. Since the symbol
does not grow fast at infinity, so there were no boundary data in the corresponding problem.
In the present paper we consider the same problem as in [16] but with symbol , where
. The approach used in this paper is more general and simple than the one used in [16]; however to get the same result are necessary more accurate estimates than the ones obtained here for the Green operator.
To construct Green operator we proposed a new method based on the integral representation for a sectionally analytic function and the theory of singular integro-differential equations with Hilbert kernel (see [16, 17]). We arrive to a boundary condition of type , where
. The aim is to find two analytic functions,
and
(a sectionally analytic function
), in the left and right complex semi-planes, respectively, such that the boundary condition is satisfied. Two conditions are necessary to solve the problem: first, the function
must satisfy the H
lder condition both in the finite points and in the vicinity of the infinite point of the contour and, second, the index of function
must be zero. In our case both conditions do fail. To overcome this difficulty, we introduce an auxiliary function such that the H
lder and zero-index conditions are fulfilled.
To state precisely the results of the present paper we give some notations. We denote ,
. Here and below
is the main branch of the complex analytic function in the complex half-plane Re
, so that
(we make a cut along the negative real axis
). Note that due to the analyticity of
for all Re
the inverse Laplace transform gives us the function which is equal to
for all
. Direct Laplace transformation
is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ4_HTML.gif)
and the inverse Laplace transformation is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ5_HTML.gif)
Weighted Lebesgue space is where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ6_HTML.gif)
for ,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ7_HTML.gif)
Now, we define the metric spaces
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ8_HTML.gif)
where , with the norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ9_HTML.gif)
where and
, with the norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ10_HTML.gif)
,
.
Now we state the main results. We introduce by formula
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ11_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ12_HTML.gif)
We define the linear functional :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ13_HTML.gif)
Theorem 1.1.
Suppose that for small the initial data
are such that the norm
is sufficiently small. Then, there exists a unique global solution
to the initial-boundary value problem (1.1). Moreover the following asymptotic is valid:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ14_HTML.gif)
for in
, where
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ15_HTML.gif)
2. Preliminaries
In subsequent consideration we shall have frequently to use certain theorems of the theory of functions of complex variable, the statements of which we now quote. The proofs may be found in all text-book of the theory. Let be smooth contour and
a function of position on it.
Definition 2.1.
The function is said to satisfy on the curve
the Hölder condition, if for two arbitrary points of this curve
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ16_HTML.gif)
where and
are positive numbers.
Theorem 2.2.
Let be a complex function, which obeys the Hölder condition for all finite
and tends to a definite limit
as
, such that for large
the following inequality holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ17_HTML.gif)
Then Cauchy type integral
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ18_HTML.gif)
constitutes a function analytic in the left and right semiplanes. Here and below these functions will be denoted by and
, respectively. These functions have the limiting values
and
at all points of imaginary axis
, on approaching the contour from the left and from the right, respectively. These limiting values are expressed by Sokhotzki-Plemelj formulae:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ19_HTML.gif)
Subtracting and adding the formula (2.4) we obtain the following two equivalent formulae:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ20_HTML.gif)
which will be frequently employed hereafter.
We consider the following linear initial-boundary value problem on half-line:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ21_HTML.gif)
Setting ,
, we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ22_HTML.gif)
where the function is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ23_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ24_HTML.gif)
for . Here and below
, where
and
are a left and right limiting values of sectionally analytic function
given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ25_HTML.gif)
where for some fixed real point ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ26_HTML.gif)
All the integrals are understood in the sense of the principal values.
Proposition 2.3.
Let . Then there exists a unique solution
for the initial-boundary value problem (2.6), which has an integral representation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ27_HTML.gif)
Proof.
In order to obtain an integral representation for solutions of the problem (2.6) we suppose that there exist a solution , which is continued by zero outside of
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ28_HTML.gif)
Let be a function of the complex variable
, which obeys the Hölder condition for all
, such that Re
. We define the operator
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ29_HTML.gif)
Using the Laplace transform we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ30_HTML.gif)
Since is analytic for all Re
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ31_HTML.gif)
Therefore, applying the Laplace transform with respect to to problem (2.6) and using (2.15) and (2.16), we obtain for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ32_HTML.gif)
We rewrite (2.17) in the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ33_HTML.gif)
with some function such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ34_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ35_HTML.gif)
Applying the Laplace transform with respect to time variable to (2.18), we find
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ36_HTML.gif)
where Re and Re
. Here, the functions
and
are the Laplace transforms for
and
with respect to time, respectively. In order to obtain an integral formula for solutions to the problem (2.6) it is necessary to know the function
. We will find the function
using the analytic properties of the function
in the right-half complex planes Re
and Re
. Equation (2.16) and the Sokhotzki-Plemelj formulae imply for Re
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ37_HTML.gif)
In view of Sokhotzki-Plemelj formulae via (2.21) the condition (2.22) can be written as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ38_HTML.gif)
where the sectionally analytic functions and
are given by Cauchy type integrals:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ39_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ40_HTML.gif)
To perform the condition (2.23) in the form of a nonhomogeneous Riemann-Hilbert problem we introduce the sectionally analytic function:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ41_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ42_HTML.gif)
Taking into account the assumed condition (2.19), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ43_HTML.gif)
Also observe that from (2.24) and (2.26) by Sokhotzki-Plemelj formulae,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ44_HTML.gif)
Substituting (2.23) and (2.28) into this equation we obtain for Re
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ45_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ46_HTML.gif)
Equation (2.30) is the boundary condition for a nonhomogeneous Riemann-Hilbert problem. It is required to find two functions for some fixed point , Re
:
, analytic in the left-half complex plane Re
and
, analytic in the right-half complex plane Re
, which satisfy on the contour Re
the relation (2.30).
Note that bearing in mind formula (2.27) we can find the unknown function , which involved in the formula (2.21), by the relation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ47_HTML.gif)
The method for solving the Riemann problem is based on the following results. The proofs may be found in [17].
Lemma 2.4.
An arbitrary function given on the contour
, satisfying the Hölder condition, can be uniquely represented in the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ48_HTML.gif)
where are the boundary values of the analytic functions
and the condition
holds. These functions are determined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ49_HTML.gif)
Lemma 2.5.
An arbitrary function given on the contour
, satisfying the Hölder condition, and having zero index,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ50_HTML.gif)
is uniquely representable as the ratio of the functions and
, constituting the boundary values of functions,
and
, analytic in the left and right complex semiplane and having in these domains no zero. These functions are determined to within an arbitrary constant factor and given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ51_HTML.gif)
In the formulations of Lemmas 2.4 and 2.5 the coefficient and the free term
of the Riemann problem are required to satisfy the H
lder condition on the contour Re
. This restriction is essential. On the other hand, it is easy to observe that both functions
and
do not have limiting value as
. So we cannot find the solution using
. The principal task now is to get an expression equivalent to the boundary value problem (2.30), such that the conditions of lemmas are satisfied. First, we introduce the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ52_HTML.gif)
where ,
, and
. We make a cut in the plane
from point
to point
through
. Owing to the manner of performing the cut the functions
and
are analytic for Re
and the function
is analytic for Re
.
We observe that the function , given on the contour Re
, satisfies the Hölder condition and
does not vanish for any Re
. Also we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ53_HTML.gif)
Therefore in accordance with Lemma 2.5 the function can be represented in the form of the ratio
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ54_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ55_HTML.gif)
From (2.37) and (2.39) we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ56_HTML.gif)
where . We note that (2.41) is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ57_HTML.gif)
Now, we return to the nonhomogeneous Riemann-Hilbert problem defined by the boundary condition (2.30). We substitute the above equation in (2.30) and add in both sides to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ58_HTML.gif)
On the other hand, by Sokhotzki-Plemelj formulae and (2.25), . Now, we substitute
from this equation in formula (2.43); then by (2.41) we arrive to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ59_HTML.gif)
In subsequent consideration we shall have to use the following property of the limiting values of a Cauchy type integral, the statement of which we now quote. The proofs may be found in [17].
Lemma 2.6.
If is a smooth closed contour and
a function that satisfies the Hölder condition on
, then the limiting values of the Cauchy type integral
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ60_HTML.gif)
also satisfy this condition.
Since satisfies on Re
the Hölder condition, on basis of Lemma 2.6 the function
also satisfies this condition. Therefore, in accordance with Lemma 2.4 it can be uniquely represented in the form of the difference of the functions
and
, constituting the boundary values of the analytic function
, given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ61_HTML.gif)
Therefore, (2.44) takes the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ62_HTML.gif)
The last relation indicates that the function , analytic in Re
, and the function
, analytic in Re
, constitute the analytic continuation of each other through the contour Re
. Consequently, they are branches of a unique analytic function in the entire plane. According to Liouville theorem this function is some arbitrary constant
. Thus, we obtain the solution of the Riemann-Hilbert problem defined by the boundary condition (2.30):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ63_HTML.gif)
Since is defined by a Cauchy type integral, with density
, we have
, as
for
. Using this property in (2.48) we get
and the limiting values for
are given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ64_HTML.gif)
Now, we proceed to find the unknown function involved in the formula (2.21) for the solution
of the problem (2.6). First, we represent
as the limiting value of analytic functions on the left-hand side complex semiplane. From (2.41) and Sokhotzki-Plemelj formulae we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ65_HTML.gif)
Now, making use of (2.49) and the above equation, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ66_HTML.gif)
Thus, by formula (2.32),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ67_HTML.gif)
We observe that is boundary value of a function analytic in the left-hand side complex semi-plane and therefore satisfies our basic assumption (2.19). Having determined the function
, bearing in mind formula (2.21) we determine the required function
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ68_HTML.gif)
Now we prove that, in accordance with last relation, the function constitutes the limiting value of an analytic function in Re
. In fact, making use of Sokhotzki-Plemelj formulae and using (2.41), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ69_HTML.gif)
Thus, the function is the limiting value of an analytic function in Re
. We note the fundamental importance of the proven fact, the solution
constitutes an analytic function in Re
, and, as a consequence, its inverse Laplace transform vanishes for all
. We now return to solution
of the problem (2.6). Taking inverse Laplace transform with respect to time and space variables, we obtain
where the function
is defined by formula (2.8). Thus, Proposition 2.3 has been proved.
Now we collect some preliminary estimates of the Green operator .
Lemma 2.7.
The following estimates are true, provided that the right-hand sides are finite:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ70_HTML.gif)
where ,
,
,
, and
and
are given by (1.13) and (1.11), respectively.
Proof.
First, we estimate the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ71_HTML.gif)
We note that , as
, and write
in the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ72_HTML.gif)
For first integral in (2.57), we obtain the estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ73_HTML.gif)
where , and for second integral we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ74_HTML.gif)
Therefore, substituting (2.58) and (2.59) in (2.57), we get for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ75_HTML.gif)
Now, we estimate function defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ76_HTML.gif)
Using (2.60), we get for the estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ77_HTML.gif)
where Re . Then, by (2.62) and Cauchy Theorem,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ78_HTML.gif)
Equations (2.63) imply that we can write in the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ79_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ80_HTML.gif)
Thus, for Re ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ81_HTML.gif)
where satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ82_HTML.gif)
In fact, we use (2.62) and inequality , where Re
Re
and
, to obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ83_HTML.gif)
Making the change of variable , (2.67) follows. Now, substituting (2.66) in (2.8), for Green function
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ84_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ85_HTML.gif)
The function defined in (2.70) satisfies the estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ86_HTML.gif)
In fact, using (2.67) we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ87_HTML.gif)
Here,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ88_HTML.gif)
We have used inequality , where
and
is some positive constant. Taking
and making the change of variables:
and
, we obtain (2.71). Now, let us split (2.69):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ89_HTML.gif)
By Fubini's theorem and Cauchy's theorem, from the first and fourth summands we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ90_HTML.gif)
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ91_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ92_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ93_HTML.gif)
Now, we show that function defined by (2.78) satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ94_HTML.gif)
In fact, using (2.62) and the inequality , where Re
and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ95_HTML.gif)
Then, taking ,
and making the change of variable
and
, we obtain (2.79). In the same way, we show that function
defined in (2.78) satisfies the inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ96_HTML.gif)
In fact, using the inequality , where Re
and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ97_HTML.gif)
Making the change of variable , we arrive to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ98_HTML.gif)
Thus, (2.81) follows. Finally, we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ99_HTML.gif)
where is given by (1.11). Making the change of variable
,
, and choosing
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ100_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ101_HTML.gif)
Now, making the change of variable and
in equation for
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ102_HTML.gif)
Therefore, (2.84) follows. Finally, using estimates (2.71), (2.79), (2.81), and (2.84), we get the asymptotic for the Green function :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ103_HTML.gif)
where is given by (1.11) and
. By last inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ104_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ105_HTML.gif)
where and
are given by (1.13) and (1.11), respectively, and
. Thus, the first estimate in Lemma 2.7 has been proved.
Now, we are going to prove the second estimate in Lemma 2.7. First, for large , using Sokhotzki-Plemelj formulae, we have for function
, defined in (2.61),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ106_HTML.gif)
Substituting last equation in (2.8), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ107_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ108_HTML.gif)
Making the change of variable we get for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ109_HTML.gif)
To estimate , we consider an extension to the function
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ110_HTML.gif)
and we use the contours
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ111_HTML.gif)
to obtain for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ112_HTML.gif)
Let us write the function , defined in (2.61), in the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ113_HTML.gif)
where Re . Then, by Cauchy Theorem, for
the second summand in last equation is zero. Thus, using (2.62) we obtain for Re
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ114_HTML.gif)
From the last inequality and (2.97) we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ115_HTML.gif)
Taking and making the change of variables
and
, in the last inequality, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ116_HTML.gif)
From (2.92) and the estimates (2.94) and (2.101) we get the estimate . Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ117_HTML.gif)
Now, for small , we are going to prove the estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ118_HTML.gif)
where . First, we rewrite the Green function
in the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ119_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ120_HTML.gif)
The contours and
are defined in (2.96) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ121_HTML.gif)
Moreover, we have extended the function as in (2.95). Making the change of variable
and using the inequality
,
, we obtain the estimate
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ122_HTML.gif)
for . Now, we estimate
. Using
, for Re
,
, and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ123_HTML.gif)
Making the change o variables
and
, into the last inequality, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ124_HTML.gif)
By (2.104) and the estimates (2.107) and (2.109) we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ125_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ126_HTML.gif)
Thus, we get (2.103) and the second estimate in Lemma 2.7 has been proved.
Let us introduce the operators
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ127_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ128_HTML.gif)
where and
are defined in (2.104). Then, the operator
can be written in the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ129_HTML.gif)
Now, we are going to prove the third estimate in Lemma 2.7,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ130_HTML.gif)
First, we estimate the operator . Making the change of variable
, we get for the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ131_HTML.gif)
Now, we make the change of variable :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ132_HTML.gif)
Integrating by parts the last equation we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ133_HTML.gif)
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ134_HTML.gif)
for , where
are defined as above. Thus, for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ135_HTML.gif)
Therefore, from the inequalities (2.116) and (2.120) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ136_HTML.gif)
We remember some well-known inequalities.
(i)Young's Inequality. Let and
, where
,
. Then, the convolution
belongs to
, where
and Young's inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ137_HTML.gif)
holds.
(ii)Minkowski's Inequality. Let and
; then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ138_HTML.gif)
(iii)Interpolation Inequality. Let with
; then
for any
, and the interpolation inequality holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ139_HTML.gif)
where and
.
(iv)Arithmetic-Geometric Mean Inequality. If and
are nonnegative, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ140_HTML.gif)
Then, by (2.121) and Young's inequality (2.122), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ141_HTML.gif)
since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ142_HTML.gif)
Finally, using the Interpolation Inequality (2.124) and the arithmetic-geometric mean inequality (2.125), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ143_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ144_HTML.gif)
Now, we estimate the operator . First, by Cauchy Theorem we get for Re
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ145_HTML.gif)
By (2.41) we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ146_HTML.gif)
Then, using (2.131) and the inequalities , where Re
and
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ147_HTML.gif)
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ148_HTML.gif)
Then, using the inequalities (2.132) and we obtain for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ149_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ150_HTML.gif)
Thus, the last estimate and (2.129) imply the third estimate in Lemma 2.7.
Now, we are going to prove the fourth estimate in Lemma 2.7. We use (2.114). First, we estimate the operator , defined in (2.112). Using the inequality
, where
, and Minkowski's inequality (2.123), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ151_HTML.gif)
Then, Young's inequality (2.122) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ152_HTML.gif)
where ,
,
, and
. Then, by the inequality (2.121) and the change of variables
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ153_HTML.gif)
Thus, , provided
. Using
, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ154_HTML.gif)
where . We note that
, since
. Substituting (2.139) in (2.137), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ155_HTML.gif)
where ,
, and
. Now, we estimate the operator
, defined in (2.113). First, we use that function
satisfies the following inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ156_HTML.gif)
Then, by the inequality , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ157_HTML.gif)
Substituting in the last inequality the estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ158_HTML.gif)
where , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ159_HTML.gif)
Then, using and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ160_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ161_HTML.gif)
where ,
and
. Finally, from estimates (2.140) and (2.146) we obtain the fourth estimate in Lemma 2.7. Then, we have proved Lemma 2.7.
Theorem 2.8.
Let the initial data be , with
. Then, for some
there exists a unique solution
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ162_HTML.gif)
to the initial boundary-value problem (1.1). Moreover, the existence time can be chosen as follows:
, where
.
3. Proof of Theorem 1.1
By the Local Existence Theorem 2.8, it follows that the global solution (if it exist) is unique. Indeed, on the contrary, we suppose that there exist two global solutions with the same initial data. And these solutions are different at some time . By virtue of the continuity of solutions with respect to time, we can find a maximal time segment
, where the solutions are equal, but for
they are different. Now, we apply the local existence theorem taking the initial time
and obtain that these solutions coincide on some interval
, which give us a contradiction with the fact that
is the maximal time of coincidence. So our main purpose in the proof of Theorem 1.1 is to show the global in time existence of solutions.
First, we note that Lemma 2.7 implies for the Green operator the inequality
Now, we show the estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ163_HTML.gif)
for all , where
,
. In fact, using the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ164_HTML.gif)
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ165_HTML.gif)
where , and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ166_HTML.gif)
Then, the estimates (3.3), (3.4), and Lemma 2.7 imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ167_HTML.gif)
where , and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ168_HTML.gif)
Now, we integrate with respect to , on the interval
, the inequalities (3.5) and (3.6). Then, we get for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ169_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ170_HTML.gif)
where , and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ171_HTML.gif)
Then, the definition of the norm on the space and the estimates (3.7), (3.8), and (3.9) imply (3.1). Now, we apply the Contraction Mapping Principle on a ball with ratio
in the space
,
, where
. Here, the constant
coincides with the one that appears in estimate (3.1). First, we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ172_HTML.gif)
where . Indeed, from the integral formula
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ173_HTML.gif)
and the estimate (3.1) (with ), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ174_HTML.gif)
since is sufficient small. Therefore, the operator
transforms a ball of ratio
into itself, in the space
. In the same way we estimate the difference of two functions
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ175_HTML.gif)
since is sufficient small. Thus,
is a contraction mapping in
. Therefore, there exists a unique solution
to the Cauchy problem (1.1). Now we can prove asymptotic formula:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ176_HTML.gif)
where . We denote
. From Lemma 2.7 we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ177_HTML.gif)
for all Also in view of the definition of the norm
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ178_HTML.gif)
By a direct calculation we have for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ179_HTML.gif)
where , provided that
, and in the same way
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ180_HTML.gif)
provided that . Also we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ181_HTML.gif)
for all By virtue of the integral equation (3.11) we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F946143/MediaObjects/13661_2010_Article_68_Equ182_HTML.gif)
All summands in the right-hand side of (3.20) are estimated by via estimates (3.17)–(3.19). Thus by (3.20) the asymptotic (3.14) is valid. Theorem 1.1 is proved.
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Árciga A, M. Asymptotics for Nonlinear Evolution Equation with Module-Fractional Derivative on a Half-Line. Bound Value Probl 2011, 946143 (2011). https://doi.org/10.1155/2011/946143
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DOI: https://doi.org/10.1155/2011/946143