An Energy Inequality and Its Applications of Nonlocal Boundary Conditions of Mixed Problem for Singular Parabolic Equations in Nonclassical Function Spaces

The aim of this paper is to establish a priori estimates of the following nonlocal boundary conditions mixed problem for parabolic equation: ∂v/∂t − a t /x2 ∂/∂x x2∂v/∂x b x, t v g x, t , v x, 0 ψ x , 0 ≤ x ≤ , v , t E t , 0 ≤ t ≤ T, ∫ 0xv x, t dx G t , 0 ≤ t ≤ . It is important to know that a priori estimates established in nonclassical function spaces is a necessary tool to prove the uniqueness of a strong solution of the studied problems.


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 > 0, T > 0, and Ω { x, t ∈ R 2 : 0 < x < , 0 < t < T}.We will determine a solution v, in Ω of the differential equation For consistency, we have 0 where and T are fixed but arbitrary positive numbers, a t and b x, t are the known fuctions satisfying the following condition.
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.
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.

Preliminares
We transform the problem with nonhomogeneous boundary conditions into a problem with homogeneous boundary conditions.For this, we introduce a new unknown function u defined by v x, t u x, t w x, t , where Then, problem becomes where

2.6
We introduce appropriate function spaces.Let L 2 Ω be the Hilbert space of square integrable functions.To problem 2.1 , 2.2 , 2.3 , 2.5 , we associate the operator A with the domain of definition satisfying 2.4 and 2.5 .The operator A is considered from E to F, where E is the banach space consisting of u ∈ L 2 Ω satisfying the boundary conditions 2.4 and 2.5 and having the finite norm: and F is the Hilbert space of vector-value function F f, ϕ having the norm where J x h x θ 2 h θ, t dθ.

A Priori Estimate and Its Consequences
Theorem 3.1.Under Condition 1, for any function v ∈ D A , one has the following a priori estimate where c is a positive constant independent of the solution v.

3.2
Integrating by parts of the second integral on the left-hand side of 3.

3.5
The standard integration by parts of the second term on the left-hand side of 3.

3.8
In the light of Cauchy inequality, certain terms of 3.8 are then majorized as follows: x ∂u ∂t dx dt, 3.9 x ∂u ∂t dx dt.

3.16
It follows by using Lemma 3.2 and 3.18 that

3.17
Therefore, by formula 3.17 and Condition 1, we obtain The operator A : E → F with domain D A has a closure A.By virtue of the uniqueness of the limit in D Ω , the identies 3.25 and 3.27 conduct to f ≡ 0.By virtue of 3.28 , 3.30 and the uniqueness of the limit in L 2 0, we conclude that ϕ ≡ 0. 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 c > 0 independent of v such that The range R A of the operator A is closed and R A R A .Consequence 3.7.A strong solution of the problem 2.2 , 2.3 , 2.4 , and 2.5 is unique and depends continuously on F f, ϕ ∈ F.