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Asymptotics for Nonlinear Evolution Equation with Module-Fractional Derivative on a Half-Line

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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:


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 [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 [212] 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 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 :


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:


for in , where and


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


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:


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.

Proposition 2.3.

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 [17].

Lemma 2.4.

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


Lemma 2.5.

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 [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


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 .

Lemma 2.7.

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 ,


where satisfies


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


we obtain


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.

Theorem 2.8.

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


we get


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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  • Analytic Function
  • Riemann Problem
  • Interpolation Inequality
  • Cauchy Type
  • Cauchy Theorem