- Research Article
- Open Access
Asymptotics for Nonlinear Evolution Equation with Module-Fractional Derivative on a Half-Line
Boundary Value Problems volume 2011, Article number: 946143 (2010)
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.
We study the local and global existence and asymptotic behavior for solutions to the initial-boundary value problem:
where , , and is the module-fractional derivative operator defined by
where is the Laplace transform for with respect to and is the Heaviside function:
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  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 , 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  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  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  but with symbol , where . The approach used in this paper is more general and simple than the one used in ; 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 Hlder 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 Hlder 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
and the inverse Laplace transformation is defined by
Weighted Lebesgue space is where
for , and
Now, we define the metric spaces
where , with the norm
where and , with the norm
Now we state the main results. We introduce by formula
We define the linear functional :
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:
for in , where and
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.
The function is said to satisfy on the curve the Hölder condition, if for two arbitrary points of this curve
where and are positive numbers.
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:
Then Cauchy type integral
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:
Subtracting and adding the formula (2.4) we obtain the following two equivalent formulae:
which will be frequently employed hereafter.
We consider the following linear initial-boundary value problem on half-line:
Setting , , we define
where the function is given by
for . Here and below , where and are a left and right limiting values of sectionally analytic function given by
where for some fixed real point ,
All the integrals are understood in the sense of the principal values.
Let . Then there exists a unique solution for the initial-boundary value problem (2.6), which has an integral representation:
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 :
Let be a function of the complex variable , which obeys the Hölder condition for all , such that Re . We define the operator by
Using the Laplace transform we get
Since is analytic for all Re , we have
Therefore, applying the Laplace transform with respect to to problem (2.6) and using (2.15) and (2.16), we obtain for
We rewrite (2.17) in the form
with some function such that
Applying the Laplace transform with respect to time variable to (2.18), we find
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
In view of Sokhotzki-Plemelj formulae via (2.21) the condition (2.22) can be written as
where the sectionally analytic functions and are given by Cauchy type integrals:
To perform the condition (2.23) in the form of a nonhomogeneous Riemann-Hilbert problem we introduce the sectionally analytic function:
Taking into account the assumed condition (2.19), we get
Also observe that from (2.24) and (2.26) by Sokhotzki-Plemelj formulae,
Substituting (2.23) and (2.28) into this equation we obtain for Re
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
The method for solving the Riemann problem is based on the following results. The proofs may be found in .
An arbitrary function given on the contour , satisfying the Hölder condition, can be uniquely represented in the form
where are the boundary values of the analytic functions and the condition holds. These functions are determined by
An arbitrary function given on the contour , satisfying the Hölder condition, and having zero index,
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
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 Hlder 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
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
Therefore in accordance with Lemma 2.5 the function can be represented in the form of the ratio
From (2.37) and (2.39) we get
where . We note that (2.41) is equivalent to
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
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
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 .
If is a smooth closed contour and a function that satisfies the Hölder condition on , then the limiting values of the Cauchy type integral
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
Therefore, (2.44) takes the form
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):
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
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
Now, making use of (2.49) and the above equation, we get
Thus, by formula (2.32),
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 :
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
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 .
The following estimates are true, provided that the right-hand sides are finite:
where , , , , and and are given by (1.13) and (1.11), respectively.
First, we estimate the function
We note that , as , and write in the form
For first integral in (2.57), we obtain the estimate
where , and for second integral we have
Therefore, substituting (2.58) and (2.59) in (2.57), we get for
Now, we estimate function defined by
Using (2.60), we get for the estimate
where Re . Then, by (2.62) and Cauchy Theorem,
Equations (2.63) imply that we can write in the form
Thus, for Re ,
In fact, we use (2.62) and inequality , where Re Re and , to obtain
Making the change of variable , (2.67) follows. Now, substituting (2.66) in (2.8), for Green function , we obtain
The function defined in (2.70) satisfies the estimate
In fact, using (2.67) we get
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):
By Fubini's theorem and Cauchy's theorem, from the first and fourth summands we obtain
Now, we show that function defined by (2.78) satisfies
In fact, using (2.62) and the inequality , where Re and , we get
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:
In fact, using the inequality , where Re and , we get
Making the change of variable , we arrive to
Thus, (2.81) follows. Finally, we show that
where is given by (1.11). Making the change of variable , , and choosing , we get
Now, making the change of variable and in equation for we obtain
Therefore, (2.84) follows. Finally, using estimates (2.71), (2.79), (2.81), and (2.84), we get the asymptotic for the Green function :
where is given by (1.11) and . By last inequality
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),
Substituting last equation in (2.8), we get
Making the change of variable we get for
To estimate , we consider an extension to the function :
and we use the contours
to obtain for
Let us write the function , defined in (2.61), in the form
where Re . Then, by Cauchy Theorem, for the second summand in last equation is zero. Thus, using (2.62) we obtain for Re
From the last inequality and (2.97) we get
Taking and making the change of variables and , in the last inequality, we obtain
From (2.92) and the estimates (2.94) and (2.101) we get the estimate . Thus,
Now, for small , we are going to prove the estimate
where . First, we rewrite the Green function in the form
The contours and are defined in (2.96) and
Moreover, we have extended the function as in (2.95). Making the change of variable and using the inequality , , we obtain the estimate or
for . Now, we estimate . Using , for Re , , and, we get
Making the change o variables and , into the last inequality, we obtain
By (2.104) and the estimates (2.107) and (2.109) we get
Thus, we get (2.103) and the second estimate in Lemma 2.7 has been proved.
Let us introduce the operators
where and are defined in (2.104). Then, the operator can be written in the form
Now, we are going to prove the third estimate in Lemma 2.7,
First, we estimate the operator . Making the change of variable , we get for the function
Now, we make the change of variable :
Integrating by parts the last equation we obtain
for , where are defined as above. Thus, for ,
Therefore, from the inequalities (2.116) and (2.120) we have
We remember some well-known inequalities.
(i)Young's Inequality. Let and , where , . Then, the convolution belongs to , where and Young's inequality
(ii)Minkowski's Inequality. Let and ; then
(iii)Interpolation Inequality. Let with ; then for any , and the interpolation inequality holds:
where and .
(iv)Arithmetic-Geometric Mean Inequality. If and are nonnegative, then
Then, by (2.121) and Young's inequality (2.122), we obtain
Finally, using the Interpolation Inequality (2.124) and the arithmetic-geometric mean inequality (2.125), we obtain
Now, we estimate the operator . First, by Cauchy Theorem we get for Re
By (2.41) we get
Then, using (2.131) and the inequalities , where Re and , and
Then, using the inequalities (2.132) and we obtain for
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
Then, Young's inequality (2.122) implies
where , , , and . Then, by the inequality (2.121) and the change of variables , we get
Thus, , provided . Using , it follows that
where . We note that , since . Substituting (2.139) in (2.137), we get
where , , and . Now, we estimate the operator , defined in (2.113). First, we use that function satisfies the following inequality:
Then, by the inequality , we obtain
Substituting in the last inequality the estimate
where , we obtain
Then, using and , we get
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.
Let the initial data be , with . Then, for some there exists a unique solution
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
for all , where , . In fact, using the inequality
where , and
Then, the estimates (3.3), (3.4), and Lemma 2.7 imply
where , and
Now, we integrate with respect to , on the interval , the inequalities (3.5) and (3.6). Then, we get for ,
where , and
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
where . Indeed, from the integral formula
and the estimate (3.1) (with ), we obtain
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 :
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:
where . We denote . From Lemma 2.7 we have
for all Also in view of the definition of the norm we have
By a direct calculation we have for
where , provided that , and in the same way
provided that . Also we have
for all By virtue of the integral equation (3.11) we get
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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