Asymptotics for Nonlinear Evolution Equation with Module-Fractional Derivative on a Half-Line
© Martín P. Árciga A. 2011
Received: 22 April 2010
Accepted: 16 June 2010
Published: 29 June 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.
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 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.
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.
where and are positive numbers.
which will be frequently employed hereafter.
All the integrals are understood in the sense of the principal values.
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).
The method for solving the Riemann problem is based on the following results. The proofs may be found in .
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 .
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 .
also satisfy this condition.
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 .
where , , , , and and are given by (1.13) and (1.11), respectively.
where and are given by (1.13) and (1.11), respectively, and . Thus, the first estimate in Lemma 2.7 has been proved.
Thus, we get (2.103) and the second estimate in Lemma 2.7 has been proved.
We remember some well-known inequalities.
where and .
Thus, the last estimate and (2.129) imply the third estimate in Lemma 2.7.
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.
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.
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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