A Priori Estimation of the Solution for Mixed Problems with Integral Condition for Singular Parabolic Equations
We prove a theorem in which we get a priori estimation of the solution for mixed problems with integral condition for singular parabolic equations. Mixed problems with nonlocal boundary conditions or with nonlocal initial conditions were studied in many works lately. Our result plays an important role in the theory of heat transmission, thermoelasticity, chemical engineering, underground water flow, and plasma physics.
The importance of problems with integral condition has been pointed out by Samarskiĭ . Mathematical modelling by evolution problems with a nonlocal constraint of the form is encountered in heat transmission theory, thermoelasticity, chemical engineering, underground water flow, and plasma physics. For background information, we refer the reader to Benouar and Yurchuk , Bouziani and Benouar , Bouziani , Cannon and van der Hoek [5–7] Ionkin and Moiceev [8, 9], Kamynin , and Yurchuk [11, 12]. Mixed problems with nonlocal boundary conditions or with nonlocal initial conditions were studied in Bouziani , Byszewski [14–16], Gasymov , Ionkin [8, 9], Lazhar , and Said and Nadia . The results and the method used here are a further elaboration of those in . We should mention here that the presence of integral term in the boundary condition can greatly complicate the application of standard functional and numerical techniques. This work can be considered as a continuation of the results in [11, 20].
We consider the following mixed problem in the rectangle :
2. A Priori Estimate
Let be the Hilbert space of all sufficiently smooth functions satisfying the second and third conditions in (2) and equipped with the norm The equality implies the following inequality:
By (5) it follows that for any .
We will use (5) for the solutions of the problem (1)-(2). For the right hand side of (1) and initial condition of from (2) we introduce the space which is consisted of the vector function with the norm Here it is assumed that
Theorem 1. For any function such that and , the following inequality holds: where .
Proof. We set with and
Consider the following equality: It can be seen that the following equalities hold:
By (12), we obtain the following equality:
Integrating by parts (and using (6)), we get
The formulas (14) imply the following:
Adding (13) and (15), we get where the function
Now, it can be shown that the following inequalities hold: The equality (16) and the inequalities (18) imply the following inequality:
Lemma 2. Let on continuous nonnegative functions , and be given, where is nondecreasing. Then from the inequality implies the inequality
The above lemma can be proved by iteration method. We omit the details.
As a consequence of (19), we obtain the following inequality: Here we have used the notation (7). The right hand side of inequality (23) does not depend on . Consequently, in the right hand side of (23) the supremum can be taken. Thus, we get the following inequality: By (1), we get the following inequality: The inequalities (24) and (25) imply inequality (9).
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
J. R. Cannon, The One-Dimensional Heat Equation, vol. 23 of Encyclopedia of Mathematics and its Applications, Addison-Wesley, Menlo Park, Calif, USA, 1984.View at: MathSciNet
A. Bouziani, “On the solvability of a class of singular parabolic equations with nonlocal boundary conditions in nonclassical function spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 30, no. 7, pp. 435–447, 2002.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
L. Byszewski, Differential and Functional-Differential Problems Together with Nonlocal Conditions, Monograph 184, Cracow University of Technology, Cracow, Poland, 1995.