1. Introduction

The integral boundary conditions for evolution problems have various applications in chemical engineering, thermoelasticity, underground water flow, and population dynamics.

Cannon was the first who drew attention to these problems with an integral one-space-variable condition [1], and their importance has been pointed out by Samarskii [2]. The existence and uniqueness of the classical solution of mixed problem combining a Dirichlet and integral condition for the equation of heat demonstrated by cannon [1] using the potential method.

Always using the potential method, Kamynin established in [3] the existence and uniqueness of the classical solution of a similar problem with a more general representation.

Subsequently, more works related to these problems with an integral one-space-variable have been published, among them, we cite the work of Benouar and Yurchuk [4], Cannon and Van Der Hoek [5, 6], Cannon-Esteva-Van Der Hoek [7], Ionkin [8], Jumarhon and McKee [9], Kartynnik [10], Lin [11], Shi [12] and Yurchuk [13]. In these works, mixed problems related to one-dimensional parabolic equations of second order combining a local condition and an integral condition was discussed. Also, by referring to the articles of Bouziani [14–16] and Bouziani and Benouar [17–19], the authors have studied mixed problems with integral conditions for some partial differential equations, specially hyperbolic equation with integral condition which has been investigated in Bouziani [20].

The present paper is devoted to the study of problems with a boundary integral two-space-variables condition for second-order hyperbolic equation.

2. Setting of the Problem

In the rectangle , with , we consider the hyperbolic equation: where the coefficient is a real-valued function belonging to such that in the rest of the paper, , , , denote strictly positive constants.

we adjoin to (1) the initial conditions the Neumann condition and the integral condition where and are known functions.

We will assume that the function and satisfy a compatibility conditions with (5), that is,

The presence of integral terms in boundary conditions can, in general, greatly complicate the application of standard functional or numerical techniques specially the integral two-space-variables condition. In order to avoid this difficulty, we introduce a technique to transfer this problem to another classically less complicated one which does not contain integral conditions. For that, we establish the following lemma.

Lemma 1. Problem (1)–(5) is equivalent to the following problem :

Proof. Let be a solution of (1)–(5), we prove that
So, by integrating (1) with respect to over and and taking into account (6) and (7), we obtain and we get The integral condition (5) has nothing to do with the coefficient . Then, (11) imposes
Let now be a solution of , then we are bound to prove that
So, by integrating (1) with respect to over and taking into account that we obtain and by integrating (1) with respect to over and taking in consideration we obtain By combining the two preceding (15) and (17) and taking into account (7), we get

3. A Priori Estimate and Its Consequences

In this paper, we prove the existence and the uniqueness of the solution of the problem (1)–(5) and of the operator equation where with domain of difinition consisting of functions such that , , , and satisfies conditions (3) and (4); the operator is considered from to , where is the Banach space consisting of all functions having a finite norm and is the Hilbert space consisting of all elements for which the norm is finite.

Theorem 2. For any function , we have the inequality where is a positive constant independent of .

Proof. Multiplying the (1) by the following : and integrating over , where ,(1) by integrating over . Consequently,
Employing integration by parts in (24), and taking into account the boundary conditions in , we obtain
Using the Cauchy’s inequality, we get
Substituting (25) and (26) into (24), we obtain
Then, using the Cauchy’s inequality and according to conditions (2), we get so, we obtain where
By virtue of Lemma  7.1 in [21] and by using it twice, we find where (2) by integrating over . Consequently,
Employing integration by parts in (33) and taking into account the boundary conditions in , we obtain
By virtue of the Cauchy’s inequality, we obtain
Substituting (34) and (35) into (33) and according to conditions (2), we get where
By virtue of Lemma  7.1 in [21] and by using it twice, we find where (3) by integrating over . Consequently,
Employing integration by parts in (40) and taking into account the boundary conditions in , we obtain
Using the Cauchy’s inequality, we get
Substituting (41) and (42) into (40), we obtain
Then, using the Cauchy’s -inequality and according to conditions (2), we get so, we obtain where
By virtue of Lemma  7.1 in [21] and by using it twice, we find where
Combining inequalities (31), (38), and (47), we obtain where
The right-hand side of (49) is independent of ; hence replacing the left-hand side by its upper bound with respect to from to , we obtain the desired inequality, where .