ROTHE METHOD FOR A MIXED PROBLEM WITH AN INTEGRAL CONDITION FOR THE TWO-DIMENSIONAL DIFFUSION EQUATION

This paper deals with an initial boundary value problem with an integral condition for the two-dimensional diffusion equation. Thanks to an appropriate transformation, the study of the given problem is reduced to that of a onedimensional problem. Existence, uniqueness, and continuous dependence upon data of a weak solution of this latter are proved by means of the Rothe method. Besides, convergence and an error estimate for a semidiscrete approximation are obtained.

The diffusion equation with an integral condition can model various physical phenomena in the context of chemical engineering, thermoelasticity, population dynamics, heat conduction processes, control theory, medical science, life sciences, and so forth (see [5,13] and the references therein).It is the reason for which such problems gained much attention in recent years, not only in engineering but also in the mathematics community.
Most of the papers dealing with problems of this type were consecrated to one-dimensional equations.The first work in this direction goes back to Cannon [6].The author, with the aid of an equivalent integral equation, proved the existence and uniqueness of the classical solution for a mixed problem with an integral condition for the homogeneous one-dimensional heat equation.
In different approaches, mixed problems for second-order one-dimensional parabolic equations which combine Dirichlet and integral conditions were investigated by Kamynin [18], Ionkin [15], Cannon and van der Hoek [9,10], Yurchuk [27], Benouar and Yurchuk [1], and Bouziani [5].With regard to mixed problems for one-dimensional heat equation with Neumann and integral conditions, Cannon et al. [8] and Cannon and Hoek [11] presented numerical schemes based on finite difference method, Shi [26] established the well-posedness of the problem in a weighted fractional Sobolev space by means of the Fourier transform and a variational formulation.For similar problems for more general parabolic equations, we refer the reader to [2,3,4] in which the author used the energy-integral method to study the solvability of the posed problems in a strong sense.
As for two-dimensional homogeneous diffusion equations with an integral condition, they have recently been treated in [7,12,13,14].
Unlike all previous works, here we will prove the solvability of problem (1.1)-(1.4)via approximation by the Rothe time-discretization method (also called method of lines) after reduction to a one-space variable problem.This method is a convenient tool for both the theoretical and the numerical analysis of the studied problem.Indeed, in addition to giving the first step towards a fully discrete approximation scheme, it provides a constructive proof of the existence of a unique exact solution to the investigated problem.
We note that since 1930, the Rothe method has been used several times to solve a relatively broad complex of evolution problems by many authors (cf., e.g., [16,17,19,20,21,22,23,24]).However, up to now, no evolution problem with an integral condition over the spatial domain has been treated with the help of this method.So, our paper can be considered as a contribution to the extension of the field of application of the aforesaid method to a new kind of problems.
The paper is organized as follows.In Section 2, we show that the investigation of problem (1.1)-(1.4)can be reduced to that of the one-dimensional problem (2.2)-(2.5)via a suitable transformation.We also give notation and assumptions on data.In Section 3, we solve the time-discretized problems corresponding to (2.2)- (2.5).Then, we derive some a priori estimates for the approximations and establish convergence and existence result for problem (2.2)-(2.5)by considering firstly the case of homogeneous boundary conditions in Section 4 and secondly the nonhomogeneous case in Section 5.

Preliminaries
Exploiting an idea due to Dehghan [13], we reformulate problem (1.1)-(1.4)by introducing a new unknown function u : (0,1) × I → R defined as follows: (2.1) Then, we have to find a function u(x,t) such that where Hence, once the solution of problem (2.2)-(2.5) is obtained, the value of p will be obtained through the following formula: provided that 1 0 µ 1 (y,t)dy = 0 and u is smooth enough so that (∂u/∂x)(1,t) makes sense.Afterwards, (2.7) will be used to find θ as the solution of problem (1.1)-(1.3)with classical boundary conditions of Neumann type, whose investigation is standard numerically as well as analytically.Thus, the study of problem (1.1)-(1.4) is simply reduced to that of problem (2.2)-(2.5).We then concentrate on this latter.
In the course of this paper, (•,•) denotes the usual scalar product in L 2 (0,1) and • the corresponding norm.We denote by V the set of all φ ∈ L 2 (0,1) which fulfil 1 0 φ(x)dx = 0. Obviously, V is a closed linear subspace of L 2 (0,1) and, consequently, it is a Hilbert space for the L 2 (0,1)-inner product.By H 2 (0,1) we denote the usual second-order Sobolev space on (0,1) with norm • H 2 (0,1) .Let X be a normed linear space.Then L 2 (I,X) and L ∞ (I,X) denote the sets of all measurable functions v : respectively.By C(I,X) and C 0,1 (I,X) we denote the sets of continuous and Lipschitz continuous mapping v : I → X, respectively, where the first one is normed by while by C 1,1 (I,X) we denote the subset of all v ∈ C 0,1 (I,X) such that dv/dt ∈ C 0,1 (I,X).Moreover, our analysis requires the use of the nonclassical function space B 1 2 (0,1) introduced by Bouziani in [4] in the following way.Let C 0 (0,1) be the space of real continuous functions with compact support in (0,1).Since such functions are Lebesgue-integrable, we can define on C 0 (0,1) the following inner product: (2.10) the inequality N. Merazga and A. Bouziani 903 holds for every v ∈ C 0 (0,1).This fact implies that C 0 (0,1) is not complete for (•,•) B , otherwise it would be so too for (•,•), which is not true.Denote by B 1 2 (0,1) the completion of C 0 (0,1) for this new inner product.Then, from (2.12), we readily see that L 2 (0,1) is a subset of B 1 2 (0,1) and, furthermore, the embedding L 2 (0,1) → B 1 2 (0,1) is continuous.Note that, by a density argument, inequality (2.12) can be extended to functions in L 2 (0,1).
In the sequel, any function (x,t) ∈ (0,1) × I → g(x,t) ∈ R is automatically identified with the associated abstract function t → g(t) defined from I into some function space on (0,1) by setting g(t) : x ∈ (0,1) → g(x,t).The strong convergence is denoted by →, while and * stand for the weak and weak * convergence, respectively.By C we denote a generic positive constant.We formulate the following assumptions which are supposed to hold throughout the paper: (A 1 ) f (t) ∈ L 2 (0,1) for each t ∈ I, and the Lipschitz condition holds for arbitrary t,t ∈ I; We look for a weak solution in the following sense.Definition 2.1.A function u : I → L 2 (0,1) is called a weak solution to problem (2.2)-(2.5)if the following conditions are satisfied: 2 (0,1)); (iii) u fulfils the initial condition (2.3) and the integral condition (2.5); (iv) let γ : (0,1) × I → R be the function Then the integral identity holds for all v ∈ L 2 (I,V).We remark that due to (i), condition (iii) has sense, and by virtue of (i), (ii), and assumption (A 2 ), each term in the integral relation (iv) is well defined.

Solvability of time-discretized problems
In order to solve problem (2.2)-(2.5)by the Rothe method, we subdivide the time interval I by points t j = jh, j = 0,...,n, where h = T/n is a step time.Then, for each n 1, problem (2.2)-(2.5)may be approximated by the following recurrent sequence of time-discretized problems.
Starting from find, successively for j = 1,...,n, functions u j : (0,1) → R such that where ), E j = E(t j ), and or stands for the first or the second derivative with respect to x, respectively.Because of the nonclassical condition (3.4), no standard method can be directly used to solve (3.2)- (3.4).Following an idea of [25], we consider the auxiliary Neumann boundary value problem for a second-order linear ordinary differential equation where w 0 = U 0 and λ j is for the moment an arbitrary but fixed real number.Since f j ∈ L 2 (0,1), the Lax-Milgram lemma implies, as it is well known, the existence and uniqueness of a solution w j ∈ H 2 (0,1) to the elliptic problem (3.5) provided that the previous function w j−1 is already known.Thus, starting with j = 1, this iterative procedure yields the following lemma.Lemma 3.1.For all n 1 and for all λ j ∈ R, the auxiliary problems (3.5), j = 1,...,n, have unique solutions w j ∈ H 2 (0,1).
To emphasize the fact that w j depends on λ j , we will write w j (•,λ j ) instead of w j .We now introduce, for each j = 1,...,n, the real function (3.6)

N. Merazga and A. Bouziani 905
We remark that w j (•,λ j ) will be a solution to problem (3.2)-(3.4)if and only if λ j is a real root of Φ j so that to establish the existence of a unique solution to (3.2)-(3.4), it is sufficient to show that Φ j admits exactly one real root.We then express w j (•,λ j ) in terms of λ j .For this, we introduce a new unknown function v j by then an easy computation shows that v j thus defined in (3.7) is a solution to the problem Consequently, v j is the superposition of v j and v j which are, respectively, solutions of the following problems (3.9) Obviously, only v j depends on λ j .Applying the "variation of parameters method," we easily obtain and substituting in (3.7), we get so that the function (3.6) can be written in the form which proves that Φ j possesses a unique root λ j ∈ R given by Thus, we have just proved the following theorem.
We can now introduce the Rothe function u (n) : I → H 2 (0,1) obtained from the functions u j by piecewise linear interpolation with respect to time as well as the step function u (n) : I → H 2 (0,1) defined as follows: The functions u (n) and u (n) are intended to be approximations of the solution of our problem (2.2)-(2.5) in some suitable function space.To confirm this fact, we derive some a priori estimates for u j and δu j .We first work with the following special case.

Case of homogeneous boundary conditions
Throughout this section, we assume that Then, for each j = 1,...,n, problem (3.2)-(3.4) is written as follows: Proof.As it will be seen later, the first estimate follows from the second one, hence we begin by this latter.
Taking, for all j = 1,...,n, the inner product in B 1 2 (0,1) of (4.2) with any φ ∈ V , we get It follows from (4.3) that so that the standard integration by parts leads to for all φ ∈ V .Consider the identity which results from (4.11) with j = 1.Performing an integration by parts, we get but assumption (A 3 ) and the first condition in (4.5) yield from which it follows that Substituting in the right-hand side of (4.12), (4.15) becomes Since δu 1 is an element of V in view of (4.4) with j = 1, the second condition in (4.5), and assumption (A 3 ), it may be employed as a test function in (4.16) to get with the aid of Cauchy-Schwarz inequality δu 1 B 1 2 (0,1) , (4.17) hence Now we take the difference of the relations (4.11) and (4.11) with j replaced by j − 1, j = 2,...,n, applied to the test function φ = δu j which is in V by virtue of (4.4) and (4.4) with j replaced by j − 1; we have Applying Cauchy-Schwarz inequality and omitting the second term in the lefthand side, we obtain f j − f j−1 B 1 2 (0,1) δu j B 1 2 (0,1) + δu j−1 B 1 2 (0,1) δu j B 1 2 (0,1) .(4.20)

N. Merazga and A. Bouziani 909
Hence, invoking assumption (A 1 ), we have so that, by an iterative procedure, we may arrive at Finally, in light of (4.18), we obtain for every j = 1,...,n.This proves (4.7) with C = lT + f C(I,B 1 2 (0,1)) + U 0 B 1 2 (0,1) .Next, we majorize u j .The application of the formula to (4.11) with φ = u j − u j−1 as test functions, j = 1,...,n, yields Omitting the first two terms in the left-hand side and using Cauchy-Schwarz inequality, we obtain So, in consideration of (4.23), we have Iterating this inequality, we may obtain Hence, the estimate (4.6) follows with and so our proof is complete.

On the diffusion equation with an integral condition
As a consequence of Lemma 4.1 and the definition of u (n) and u (n) , we obtain the following corollary.Corollary 4.2.For all n 1, the functions u (n) and u (n) satisfy the estimates C for a.e.t ∈ I, (4.31) For estimate (4.32), it suffices to see that we have and consequently, Hence, applying estimate (4.7), we get (4.32) with Finally, using the inequality Then, for all v ∈ L 2 (I,V), the variational equation (4.11) may be written in terms of u (n) , u (n) , and f (n) as follows: for a.e.t ∈ I. (4.40) Integrating this formula over I, we obtain the following approximation: scheme and we propose to establish the convergence of it to the weak formulation of problem (2.2)-(2.5),given by (2.15).The results of Corollary 4.2 are the basis for the following convergence assertions for the Rothe approximations.
Actually, we may state the following uniform convergence assertion.
Theorem 4.6.The sequence {u (n) } n converges in the norm of the space C(I,B 1 2 (0, 1)) to the solution u of (2.2)- (2.5), and the error estimate takes place with some positive constant.

.69)
This implies that {u (n) } n is a Cauchy sequence in the Banach space C(I,B 1 2 (0,1)), and hence it converges in the norm of this latter to some function which is not other than u.Besides, passing to the limit m → ∞ in (4.69), we obtain the desired error estimate with C = 2 T(C 2  1 + C 2 l) and h = h n , which finishes the proof.Now, we present some additional properties of the obtained solution.
At the end of this subsection, we summarize all the obtained results into the following theorem.
Moreover, u depends continuously on the right-hand side of (2.2) and on the initial function.

Theorem 4 . 4 .
The limit function u is the unique weak solution to problem (2.2)-(2.5) in the case of(4.1)  in the sense of Definition 2.1.