ROTHE TIME-DISCRETIZATION METHOD FOR THE SEMILINEAR HEAT EQUATION SUBJECT TO A NONLOCAL BOUNDARY CONDITION

This paper is devoted to prove, in a nonclassical function space, the weak solvability of a mixed problem which combines a Neumann condition and an integral boundary condition for the semilinear one-dimensional heat equation. The investigation is made by means of approximation by the Rothe method which is based on a semidiscretization of the given problem with respect to the time variable.


Introduction
In our earlier work [5], an investigation was made for an initial-boundary value problem with an integral condition for the two-dimensional diffusion equation.There, a suitable transformation has allowed us to bring the considered problem back to an equivalent problem of the following form: ∂u ∂t − ∂ 2 u ∂x 2 = f (x,t), (x,t) ∈ (0,1) × [0,T], u(x,0) = U 0 (x), x ∈ (0,1), whose weak solvability was then proved with the help of the Rothe time-discretization method.
In the present paper, we consider a generalization of problem (1.1), namely the problem of finding a function v = v(x,t) which obeys, in a weak sense, the semilinear diffusion equation ∂v ∂t − ∂ 2 v ∂x 2 = f(x,t,v), (x,t) ∈ (0,1) × [0,T], (1.2) subject to the initial condition v(x,0) = V 0 (x), x ∈ [0,1], (1.3) the Neumann condition ∂v ∂x (0,t) = g(t), t ∈ [0,T], (1.4) and the integral boundary condition where f, V 0 , g, and E are given functions, and T is a positive constant.The method used here to investigate problem (1.2)-(1.5) is the same as in [5], the socalled "Rothe method."However, the presence of the semilinearity in (1.2) complicates the process of derivating the necessary a priori estimates and proving the convergence of the method.Moreover, we follow in Section 5 a slightly different way which is simpler and shorter than the one in [5].
It is interesting to note that problem (1.2)-(1.5)has, like (1.1), many practical interpretations in the context of chemical engineering, thermoelasticity, heat conduction theory, population dynamics, and so forth (see the references in [5]).
The plan of the paper is as follows.In Section 2, notations, assumptions on data, and some useful results are given before stating the precise sense of the required solution as well as the main result of the paper.In Section 3, a semidiscretization in time of problem (1.8)- (1.11) is performed to construct approximate solutions, the so-called "Rothe approximations."Some necessary a priori estimates for these approximations are derived in Section 4, and then used, in Section 5, to establish a convergence and existence result for the problem under study.
In addition to the standard functional spaces of the types C(I,X), C 0,1 (I,X), L 2 (I,X), and L ∞ (I,X) of continuous, Lipschitz-continuous, L 2 -Bochner integrable, and essentially bounded functions from I into a Banach space X, respectively (see, e.g., [4]), our analysis requires also the use of the nonclassical function space B 1 2 (0,1) introduced by the second author (see, e.g., [1,2]) as the completion of the space C 0 (0,1) of real continuous functions with compact support in (0,1) with respect to the inner product where then, the inequality holds for every v ∈ L 2 (0,1), and hence the embedding L 2 (0,1) → B 1 2 (0,1) is continuous.
Strong and weak convergence are denoted by → or , respectively, and the symbol c will stand for generic positive constants which may be different in the same discussion.
At several places, we will use the following continuous and discrete forms of Gronwall lemma.
Lemma Proof.The proof of assertion (i) is the same as in [3,Lemma 1.3.19].As for assertion (ii), it suffices to see that from our hypothesis, the following estimate follows: (2.11) Hence, using the elementary inequality 1 + t e t , for all t ∈ R + , we have a i Ae B(i−1)h , which was to be proved.
takes place for all φ ∈ V and a.e.t ∈ I.
We remark that the fulfillment of the integral condition (1.11) is included in the fact that u(t) ∈ V for a.e.t ∈ I.
To close this section, we announce the main result of the paper.

Theorem 2.3. Under assumptions (H1)-(H3), problem (1.8)-(1.11) admits a unique weak solution u in the sense of Definition 2.2 that depends continuously on the right-hand
side f and the initial function U 0 .Moreover, the following convergence properties hold: as n tends to infinity, where {u (n) } n is the sequence of Rothe approximations defined in (3.7).
The proof of this result will be carried out along the following sections.

Rothe approximations
Let n be a positive integer.Following the idea of Rothe, we solve the recurrent system of time-discretized problems: successively for j = 1,...,n, commencing with the initial value u 0 = U 0 , where t j = jh, h = T/n, and For the functions u j which can be viewed as backward finite difference approximations of u(t j ,•), we have the following result.
Proof.Similarly as in [5], the proof consists of the following two steps.
Step 1.We first look for the functions w j (x) = w j (x;λ) which solve the associated classical Neumann boundary value problems successively for j = 1,...,n, where F j (x) := f (x,t j ,w j−1 ) + (1/h)w j−1 (x), w 0 = U 0 and λ is a real parameter.Since, according to assumptions (H1) and (H2), ), the Lax-Milgram lemma guarantees the existence and uniqueness of a strong solution Step by step, each w j is then uniquely determined in terms of U 0 , w 1 ,...,w j−1 .Thus, for all n 1 and all λ ∈ R, the auxiliary problems (3.4) j , j = 1,...,n, have unique solutions w j ∈ H 2 (0,1).
Step 2. Now, let us show that for all j = 1,...,n, the parameter λ can be selected in a suitable way such that the corresponding function w j (•;λ) is exactly a solution of problem (3.1) j -(3.3) j .Obviously, this happens if and only if λ is a root of the real function Φ j (λ) defined by Φ j (λ) := From the superposition principle, we have that where w j (•;0) is the solution (uniquely determined) to problem (3.4) j for λ = 0 and χ is the (unique) solution to the following problem: One can readily find that χ is given by so that, replacing into (3.5)j , this yields that is, which shows that for all h > 0, Φ j admits a unique root λ = λ j ∈ R, namely λ j = −(1/h) 1 0 w j (x;0)dx.Hence, problem (3.1) j -(3.3) j is uniquely solvable for all n 1 and all j = 1,...,n.Therefore, Theorem 3.1 has been proved.Now, for all n 1, we introduce the Rothe approximation u (n) : and the corresponding step function u (n) : I → H 2 (0,1) ∩ V is defined as follows: We expect that the limit lim n→∞ u (n) = u exists in a suitable sense, and that is precisely the desired weak solution to our problem (1.8)- (1.11).The establishment of this fact requires some a priori estimates whose derivation is the subject of the following section.

A priori estimates for the approximations
Lemma 4.1.There exist c > 0 such that for all n 1, the solutions u j of the time-discretized problems (3.1) j -(3.3) j , j = 1,...,n, obey the estimates u j c, (4.1) Proof.The key point to establish these estimates is the derivation of a nonstandard variational formulation of problems (3.1) j -(3.3) j .To this aim, we take, for all j = 1,...,n, the inner product in B 1 2 (0,1) of (3.1) j with any function φ from the space V defined in (2.1) to get But from (3.2) j we have then since φ ∈ V .Substituting in (4.3), this yields the required variational form: δu j ,φ B 1 2 (0,1) + u j ,φ = f j ,φ B 1 2 (0,1) , (4.4) j which gives for j = 1 that Integrating by parts the second term in the right-hand side of (4.6), we have but, due to assumption (H3) 1 , we note that N. Merazga and A. Bouziani 9 whence x φ dx, (4.9) so that (4.6) becomes , ∀φ ∈ V. (4.10) Testing this last equality with φ = δu 1 = (u 1 − U 0 )/h which is clearly an element of V because of (3.3) 1 and assumption (H3) 2 , we derive with the help of Cauchy-Schwarz inequality and then Next, subtracting (4.4) j−1 from (4.4) j ( j = 2,...,n) and putting φ = δu j which belongs to V in view of (3.3) j−1 and (3.3) j , we estimate which implies that then, iterating we may arrive at But owing to assumption (H1), we have for all i 2 that (4.18) To dominate the right-hand side in (4.18), we need to estimate the term u i B 1 2 .For this, we take φ = u i in (4.4) i , i = 1,...,n, and get from where we derive and from this recurrent inequality, we successively estimate Invoking assumption (H1), we have for all k 1 that where M := max t∈I f (t,0) B 1 2 < +∞.Substituting (4.22) in the previous inequality, we get from where it comes due to the discrete Gronwall's lemma that (4.26) Combining (4.13), (4.16), and the last inequality, we have hence, applying Gronwall's lemma in discrete form again, we get e l( j−1)h .(4.28) Thus, estimate (4.2) is proved for c = c 3 with

.29)
Next, to derive estimate (4.1), we insert φ = u j − u j−1 in (4.4) j and apply the identity to get Ignoring the first two terms in the left-hand side, we obtain whence, using (4.22), (4.25), and (4.2), So, by an iterative procedure, we get from where estimate (4.1) follows with c = c 4 , where and so the proof is complete.
If we extend, for all n 1, the function u (n) defined on I to the interval [−T/n,0) by setting we can state the following corollary.
Corollary 4.2.For all n 1, the functions u (n) and u (n) satisfy the estimates ) Proof.Both estimates (4.37) follow immediately from (4.1) with the same constant c = c 4 .
On the other hand, invoking the identity estimate (4.38) is seen to be an easy consequence of estimate (4.2) with c = c 3 .Next, observing that we have we can write hence, in view of (4.2), we get the required estimates (4.39) and (4.40) with c = c 3 T.

Existence, uniqueness, and convergence of the method
Let us define, for all n 1, the abstract step function f (n) : N. Merazga and A. Bouziani 13 Then, the variational equations (4.4) j may be rewritten in the form (5.1) n for all φ ∈ V and t ∈ (0,T].
It is convenient to present now a basic convergence statement.
Theorem 5.1.The sequence {u (n) } n converges in the norm of the space C(I,B 1 2 (0,1)) to some function u ∈ C(I,B 1 2 (0,1)) and the error estimate holds for all n 1.
Merazga and A. Bouziani 15 t ∈ (t k−1 ,t k ] ∩ (t i−1 ,t i ], hence from assumption (H1), it follows that .11) Now, let t be arbitrary but fixed in (0,T], then there exist two integers k and i corresponding to the subdivision of I into n and m subintervals, respectively, such that N. T/l 2 and c 11 := 2T(c 9 /l 2 + c 6 ).Accordingly,