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. 1. Introduction The importance of problems with integral condition has been pointed out by Samarski？ [1]. 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 [2], Bouziani and Benouar [3], Bouziani [4], Cannon and van der Hoek [5–7] Ionkin and Moiceev [8, 9], Kamynin [10], and Yurchuk [11, 12]. Mixed problems with nonlocal boundary conditions or with nonlocal initial conditions were studied in Bouziani [13], Byszewski [14–16], Gasymov [17], Ionkin [8, 9], Lazhar [18], and Said and Nadia [19]. The results and the method used here are a further elaboration of those in [2]. 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 with . 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.