- Research Article
- Open Access
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.
- Analytic Function
- Riemann Problem
- Interpolation Inequality
- Cauchy Type
- Cauchy Theorem
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.
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 .
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.
Thus, we get (2.103) and the second estimate in Lemma 2.7 has been proved.
We remember some well-known inequalities.
Thus, the last estimate and (2.129) imply the third estimate in Lemma 2.7.
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.
- Hayashi N, Kaikina EI, Naumkin PI, Shishmarev IA: Asymptotics for Dissipative Nonlinear Equations, Lecture Notes in Mathematics. Volume 1884. Springer, Berlin, Germany; 2006:xii+557.Google Scholar
- Alber HD, Zhu P: Global solutions to an initial boundary value problem for the Mullins equation. Journal of Partial Differential Equations 2007, 20(1):30–44.MathSciNetMATHGoogle Scholar
- Benchohra M, Hamani S: Nonlinear boundary value problems for differential inclusions with Caputo fractional derivative. Topological Methods in Nonlinear Analysis 2008, 32(1):115–130.MathSciNetMATHGoogle Scholar
- Fokas AS: The Davey-Stewartson equation on the half-plane. Communications in Mathematical Physics 2009, 289(3):957–993. 10.1007/s00220-009-0809-1View ArticleMathSciNetMATHGoogle Scholar
- Fu Y, Li Y: Initial boundary value problem for generalized 2D complex Ginzburg-Landau equation. Journal of Partial Differential Equations 2007, 20(1):65–70.MathSciNetMATHGoogle Scholar
- Friedrich H: Initial boundary value problems for Einstein's field equations and geometric uniqueness. General Relativity and Gravitation 2009, 41(9):1947–1966. 10.1007/s10714-009-0800-3View ArticleMathSciNetMATHGoogle Scholar
- Shao Z-Q: Global existence of classical solutions to the mixed initial-boundary value problem for quasilinear hyperbolic systems of diagonal form with large BV data. Journal of Mathematical Analysis and Applications 2009, 360(2):398–411. 10.1016/j.jmaa.2009.06.066View ArticleMathSciNetMATHGoogle Scholar
- Caballero Mena J, Harjani J, Sadarangani K: Existence and unqiueness of positive and nondecreasing solutions for a class of singular fractional boundary value problems. Boundary Value Problems 2009, 2009:-10.Google Scholar
- Belmekki M, Nieto JJ, Rodríguez-López R: Existence of periodic solution for a nonlinear fractional differential equation. Boundary Value Problems 2009, 2009:-18.Google Scholar
- Ahmad B, Nieto JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Boundary Value Problems 2009, 2009:-11.Google Scholar
- Ahmad B, Otero-Espinar V: Existence of solutions for fractional differential inclusions with antiperiodic boundary conditions. Boundary Value Problems 2009, 2009:-11.Google Scholar
- Benchohra M, Cabada A, Seba D: An existence result for nonlinear fractional differential equations on Banach spaces. Boundary Value Problems 2009, 2009:-11.Google Scholar
- Hayashi N, Kaikina EI: Nonlinear Theory of Pseudodifferential Equations on a Half-Line, North-Holland Mathematics Studies. Volume 194. Elsevier Science B.V., Amsterdam, The Netherlands; 2004:xx+319.Google Scholar
- Kaikina EI: Subcritical pseudodifferential equation on a half-line with nonanalytic symbol. Differential and Integral Equations 2005, 18(12):1341–1370.MathSciNetMATHGoogle Scholar
- Kaikina EI: Nonlinear pseudoparabolic type equations on a half-line with large initial data. Nonlinear Analysis. Theory, Methods & Applications 2007, 67(10):2839–2858. 10.1016/j.na.2006.09.044View ArticleMathSciNetMATHGoogle Scholar
- Kaikina EI: Pseudodifferential operator with a nonanalytic symbol on a half-line. Journal of Mathematical Physics 2007, 48(11):1341–1370.View ArticleMathSciNetGoogle Scholar
- Gakhov FD: Boundary value problems. Pergamon Press, Oxford, UK; 1966:xix+561.MATHGoogle Scholar
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