1. Introduction

In this paper, we deal with a class of parabolic equations with time- and space-variable characteristics, with a nonlocal boundary condition. The precise statement of the problem is a follows: let , and . We will determine a solution , in of the differential equation satisfying the initial condition the classical condition and the integral condition For consistency, we have where and are fixed but arbitrary positive numbers, () and () are the known fuctions satisfying the following condition.

Condition 1. For and , we assume that (i), (ii), (iii) The notion of nonlocal condition has been introduced to extend the study of the classical initial value problems and it is more precise for describing natural phenomena than the classical condition since more information is taken into account, thereby decreasing the negative effects incurred by a possibly erroneous single measurement taken at the initial value. The importance of nonlocal conditions in many applications is discussed in [1, 2].

It can be a part in the contribution of the development of a priori estimates method for solving such problems. The questions related to these problems are so miscellaneous that the elaboration of a general theory is still premature. Therefore, the investigation of these problems requires at every time a separate study.

This work can be considered as a continuation of the results of Yurchuk [3], Benouar and Yurchuk[4], Bouziani [5–7], Bouziani and Benouar [8], Djibibe et al. [9], and Djibibe and Tcharie [10]. Our results generalize and deepen ones from corresponding work in [11, 12].

We should mention here that the presence of an integral term in the boundary condition can greatly complicate the application of standard functional and numerical techniques.

This paper is organized as follows. After this introduction, in Section 2, we present the preliminaries. Finally, in Section 3, we establish an energy inaquality and give its several applications.

2. Preliminares

We transform the problem with nonhomogeneous boundary conditions into a problem with homogeneous boundary conditions. For this, we introduce a new unknown function defined by , where Then, problem becomes where We introduce appropriate function spaces. Let be the Hilbert space of square integrable functions. To problem (2.1), (2.2), (2.3), (2.5), we associate the operator with the domain of definition satisfying (2.4) and (2.5). The operator is considered from to , where is the banach space consisting of satisfying the boundary conditions (2.4) and (2.5) and having the finite norm: and is the Hilbert space of vector-value function having the norm where .

3. A Priori Estimate and Its Consequences

Theorem 3.1. Under Condition 1, for any function , one has the following a priori estimate where is a positive constant independent of the solution .

Proof. Firstly, applying operator to (2.1), multiplying the obtained result with , and integrating over , oberve that Integrating by parts of the second integral on the left-hand side of (3.2), we get Substituting (3.3) into (3.2), we get In the second time, multiplying the equality (2.1) with , and integrating the obtained equality over , we get The standard integration by parts of the second term on the left-hand side of (3.5), leads to Substituting (3.6) into (3.5), we get Finally, adding (3.4) to (3.7), we have In the light of Cauchy inequality, certain terms of (3.8) are then majorized as follows: Combining the inequalities (3.9), (3.10), (3.11) with (3.8), choosing which that and , we get where

Lemma 3.2. For , the following inequalities hold: It follows by using Lemma 3.2 and (3.18) that Therefore, by formula (3.17) and Condition 1, we obtain where Eliminating the last term on the right-hand side of inequality (3.18). To this end, using Gronwall's lemma, it follows that where .
The right-hand side of (3.20) is independent of , hence, replacing the left-hand side by the upper bound with respect to , We get where . This completes the proof of Theorem 3.1.

Lemma 3.3. The operator with domain has a closure .

Proof of Lemma 3.2. Suppose that is a sequence such that we must show that and . Equality (3.22) implies that By virtue of the condition of derivation of in , we get Then from equality (3.23) it follows that therefore By virtue of the uniqueness of the limit in , the identies (3.25) and (3.27) conduct to .
By analogy, from (3.23), we get We see via (3.22) and the obvious inequality that By virtue of (3.28), (3.30) and the uniqueness of the limit in we conclude that .

Definition 3.4. A solution of the equation is called a strong solution of problem (2.2), (2.3), (2.4), and (2.5).

Consequence 3.5. Under the conditions of Theorem 3.1, there is a constant independent of such that

Consequence 3.6. The range of the operator is closed and .

Consequence 3.7. A strong solution of the problem (2.2), (2.3), (2.4), and (2.5) is unique and depends continuously on .