ON THE WEAK SOLUTION OF A THREE-POINT BOUNDARY VALUE PROBLEM FOR A CLASS OF PARABOLIC EQUATIONS WITH ENERGY SPECIFICATION

This paper deals with weak solution in weighted Sobolev spaces, of three-point boundary value problems which combine Dirichlet and integral conditions, for linear and quasilinear parabolic equations in a domain with curved lateral boundaries. We, firstly, prove the existence, uniqueness, and continuous dependence of the solution for the linear equation. Next, analogous results are established for the quasilinear problem, using an iterative process based on results obtained for the linear problem.


Introduction
Many physical phenomena are modeled by one-dimensional second-order parabolic equation which involves nonlocal boundary condition of the form b 0 θ(x,t)dx = E(t), (1.1) the so-called energy specification, where b ∈]0, 1] is a constant, θ(x,t) is an unknown function, and E(t) is a given function.
For heat conduction theory, condition (1.1) represents the internal energy content of a portion of the conductor [3,9,11,16,17].For diffusion processes, the condition is equivalent to the specification of mass in a portion of the domain of diffusion [3,10,12].We note that such problems have other important applications, for instance, in thermoelasticity [15,18], in electrochemistry [14], and in medical science [13].
In this paper, we prove in weighted Sobolev spaces the existence, uniqueness, and continuous dependence of the weak solution of three-point boundary value problem for a class of linear and quasilinear parabolic equations with nonlocal condition in a domain with curved boundaries varying with respect to the time.

Parabolic equations with energy specification
We will first investigate the linear case.Particular cases of it have been treated by several authors; most of the works were directed to strongly generalized solution for two-point boundary value problems [1,2,3,4,5,6,7,19] and to the classical solutions of the heat equation [8,9,10,11,12,16,17].In contrast to previous papers, we consider a weak solution by using a functional analysis method based on a priori estimates.Then, we investigate the quasilinear problem by combining an iterative process with results established for the linear case.
The outline of the paper is as follows.In Section 2, we study the linear problem.In Section 2.1, we give the statement of the problem, the basic assumptions, and some function spaces needed in the remainder of the work.Section 2.2 is devoted to establishing the existence of the solution.The uniqueness and continuous dependence with respect to the data are proved in Section 2.3.In Section 3, analogous investigation for the quasilinear problem is considered.
576 Parabolic equations with energy specification Assumption 2.1.For all (x,t) ∈ Ω, we assume that (2.13) Here and in the rest of the paper c i are positive constants.We now introduce some function spaces which are related to the study.By L 2 (0,1) we represent the usual space of Lebesgue square integrable functions on (0,1) whose scalar product and norm will be denoted by (•,•) L 2 (0,1) and and the associated norm u L 2 ρ (0,1) = ρu L 2 (0,1) , (2.15) where ρ(x) is a continuous function defined by (2.21)

Existence of the solution.
First, we make precisely the concept of the solution of problem (2.9), (2.10), and (2.11) we are considering in this paper.For this, we take a function v(x,t) ∈ V , the space of functions belonging to C 1 (Ω), which satisfies the following conditions: , for all t ∈ I.We now consider the inner product in L 2 (I,L 2 (0,1)) of (2.9) and the operator (2.24) We integrate by parts the first two terms on the left-hand side of (2.24).To do this, we assume that u,v In light of the above assumptions, we have .
(2.33) Substituting (2.33) into (2.32) and taking into account Assumption 2.1, we get by choosing ε 1 = 2, ε 2 = ε 3 = 3,ε 4 = 4, and dτ. (2.34) In light of the elementary inequality dt, (2.35) the last estimate becomes (2.37)Therefore, from Gronwall's lemma, we come to the conclusion that (2.38) If we omit the first term on the left-hand side of (2.38) and integrate the result over I, it yields On the other hand, if in the left-hand side of (2.38) we take the upper bound with respect to t, from 0 to T, since the right-hand side of the inequality does not depend on t, we obtain (2.40) Thus we have proved the following theorem.
From estimates (2.41), it follows immediately that {u n } n is a Cauchy sequence.Thus we have the following corollary.
Corollary 2.7.Under the assumptions of Theorem 2.6, the sequence {u n } n converges to u in the following sense:

.44)
We must prove that the limit function u(x,t) is a solution of problem (2.9), (2.10), and (2.11) in the sense of Definition 2.5.For this purpose, we consider Abdelfatah Bouziani 583 the weak formulation of problem (2.29) (2.45)However, u n = (u n − u) + u, u 0 n = (u 0 n − u 0 ) + u 0 , and f n = ( f n − f ) + f , then it follows from the last identity that In light of the Schwarz inequality and inequalities (2.21), we get the following estimates: Therefore, if we pass to the limit in equality (2.46), by taking into account the limit relations (2.42) and (2.44), and estimates (2.47), we conclude that u satisfies equality (2.27).On the other hand, it follows from Corollary 2.7 that t 0 u(x,t)dx is in C(Ω); from which we deduce that u(1,t) = 0 almost everywhere.

Parabolic equations with energy specification
It remains to prove that u satisfies the initial condition.Let (2.48) Passing to the limit as n → +∞ in the above inequality, by taking into account (2.43) and (2.28b), we get u(x,0) = u 0 (x).
We have thus proved the following theorem.
Theorem 2.8.If u 0 belongs to H 1 ρ (0,1) and f to L 2 (I,L 2 ρ (0,1)), then there exists a weak solution u of problem (2.9), (2.10), and (2.11) possessing the following properties: (2.49) Continuous dependence and uniqueness.We first establish the continuous dependence of the solution with respect to the data.The uniqueness then follows directly.
Corollary 2.10.Under the assumptions of Theorem 2.8, the weak solution of problem (2.9), (2.10) θ(ξ,t)dξ = E(t), 0 < τ < T. ( Here, we conserve the notations given in Section 2. As in Section 2.1, we reduce problem (3.1) to the following: Consider now the following auxiliary problem: Section 2 implies that problem (3.3) admits a unique weak solution that depends continuously with respect to the initial condition.Thus it remains to establish the proof for problem (3.4).For this purpose, we employ the following iteration procedure.
Let ω (0) = 0 and let {ω (n) } n be defined as follows: if ω (n−1) is given, then solve the following problem: Theorem 2.8 implies, for fixed n, that each of problems (3.5) possesses a unique solution ω (n) (x,t).Taking its difference for n = k and n = k + 1, it follows that where (3.7)

Uniqueness of the solution.
In this section, we first establish an a priori estimate.The uniqueness of the solution is then a direct corollary of it.
Theorem 3.3.Under the hypotheses of Theorem 3.1, assume that ω 1 and ω 2 are two solutions in B of problem (3.4).Then and 578 Parabolic equations with energy specification b 0

Theorem 3 . 2 .
Let the hypotheses of Theorem 3.1 hold.If then problem (3.4) admits a weak solution in the space B.
• L 2 (0,1) , respectively.By C 0 (0,b) we denote the space of continuous functions with compact support in (0,b).Let H be a Hilbert space with the norm u H and let u : I → H be an abstract function.By u(•,t) H we denote the norm of the element u(•,t) ∈ H at a fixed t.The space C(I,H) is the set of all continuous functions u : I → H with u C(I,H) = max t∈[I] u(•,t) H < ∞, and L 2 (I,H) is the set of the measurable abstract functions u such that u L 2 (I,H) =