SOLUTION TO A TRANSMISSION PROBLEM FOR QUASILINEAR PSEUDOPARABOLIC EQUATIONS BY THE ROTHE METHOD

In this paper, we deal with a transmission problem for a class of quasilinear pseudoparabolic equations. Existence, uniqueness and continuous dependence of the solution upon the data are obtained via the Rothe method. Moreover, the conver- gence of the method and an error estimate of the approximations are established.


Introduction
Let Ω be a bounded open domain in the space R N of points x = (x 1 , ..., x N ) with the Lipschitz boundary ∂Ω, such that where Γ 0 and Γ 1 are open complementary parts, each consisting of an integer number of parts.Assume that Ω consists of M subdomains Ω k , 1 ≤ k ≤ M, (see fig. 1 ), with respective boundaries ∂Ω k .
We first introduce some notations.Let for µ = 0, 1.Moreover, let for k = 1, ..., M, One should note that ∈ N k iff k ∈ N and hence Γ k, = Γ ,k , 1 ≤ k, ≤ M. Then we consider the following problem: Determine u k = u k (x, t) (k = 1, ..., M) , x ∈ Ω k , t ∈ I = (0, T ] , which obey the respective quasilinear third order pseudoparabolic equations where along with the initial conditions as well as with transmission conditions where ∂u k ∂ϑ A is the conormal derivative defined by: a k pq (x) with cos ϑ k , x p denotes the p − th component of the outward unit normal vector ϑ k to ∂Ω k , k stands for the superscript in u k , f k and ϑ k , not an exponent.Equation (1.1) can be classified as a pseudoparabolic equation because of its close link with the corresponding parabolic equation.In fact, in several cases, the solution of parabolic problem can be obtained as a limit of solutions to the corresponding problem for (1.1) when α → 0 [28].It can be also classified as a hyperbolic equation with a dominate derivative [4].
Particular cases of problem (1.1)- (1.4) arise in various physical phenomena, for instance, in the theory of seepage of homogeneous liquids in fissured rocks [2,8,9], in the nonsteady flows of second order fluids [28], in the diffusion of imprisoned resonant radiation through a gas [20,21,27] which has applications in the analysis of certain laser systems [24] and in the modelling of the heat conduction involving a thermodynamic temperature θ = u−α∆u along with a conductive temperature u, see [7].An important particular case of problem (1.1)-(1.4) is which related to the Benjamin-Bona-Mahony equation proposed in [3].Taking into account dissipative phenomena, equation (1.5) is modified to the so-called Let us cite some interesting papers dealing with transmission problems.The first of them is that of Gelfend [12], who attracted the attention on these problems, by showing their motivation.In [23], von Petersdorff used a boundary integral method to study a transmission problem for the Helmholtz equation in a number of adjacent Lipschitz domains in R n , n ≥ 2, on the boundaries of which inhomogeneous Dirichlet, Neumann or transition conditions are imposed.Gaiduk [10], considered a linear problem about transverse vibrations of a uniform rectangular viscoelastic plate with supported boundaries caused by an impact.By means of the contour integral method, due to Rasulov [25], in combination with the method of separation of variables, it is shown the solvability and properties of the solution.In [15], Kačur-van Keer established a numerical solution for a transmission of linear parabolic problem, which is encountered in the context of transient temperature distribution in composite media consisting of several regions in contact, by applying a Rothe-Galerkin finite element method.Along a different line, transmission problems for parabolic-hyperbolic equations were considered by Ostrovsky [22], Ladyžhenskaya [18], Korzyuk [13], Lions [17] and Bouziani [5].
In this paper, we present the Rothe time-discretisation method, as a suitable method for both theoretical and numerical analysis of problem (1.1)- (1.4).Actually, in addition to providing the first step towards a fully discrete approximation scheme, it gives a constructive proof of the existence, uniqueness and continuous dependence of the solution upon the data.
Let us mention that the present work can be considered as a continuation of the previous works of the authors [6,19], where linear single pseudoparabolic equations were studied.It can also be viewed as a companion of paper [15] by Kačur and van Keer.
An outline of the paper is as follows.In Section 2, we give the basic assumptions, notations and the appropriate function spaces.We also recall some auxiliary results used in the rest of the paper.The variational formulation of problem (1.1)-(1.4)as well as the concept of the solution we are considering and the solvability of the time discretized problem corresponding to (1.1)-(1.4)are given in Section 3. In Section 4, we establish some a priori estimates for the discretized problem, EJQTDE, 2007 No. 14, p. 4 while convergence results and error estimate are given in Section 5. Section 6 is devoted to the existence, uniqueness and continuous dependence of the solution upon the data of problem (1.1)- (1.4).Finally, we establish in Section 7 the error estimate.

Preliminaries
Let H 1 (Ω k ) be the first order Sobolev space on Ω k with scalar product (•, •) 1,Ω k and corresponding norm • 1,Ω k , and let (•, •) 0,Ω k and • 0,Ω k be the scalar product and corresponding norm respectively in L 2 (Ω k ).Let the space of functions defined by: where The space V is equipped with the norm • 1,Ω , namely We identify v ∈ V with a function v : Ω → R, for which v| Ω k = v k , (k = 1, ..., M) .Similarly, we introduce the product space ) equipped with the scalar product and the associated norm respectively.Now, we state the following hypotheses which are assumed to hold for k = 1, ..., M : ) and fulfills the Lipschitz condition: where with a k pq satisfy assumptions A1-A2, then the form a (u, v) fulfills the following properties: There exists a sufficiently large constant β 0 (≥ 1) such that a (v, v) ≥ β 0 v 2 1,Ω , ∀u ∈ V. Throughout, we will identify any function (x, t) ∈ Ω × I → g(x, t) ∈ R with the associated abstract function t → g(t) defined from I into certain function space on Ω by setting g(t) : x ∈ Ω → g(x, t).Moreover, we will use the standard functional spaces L 2 (I, H), C(I, H), L ∞ (I, H) and Lip(I, H), where H is a Banach space.For their properties, we refer the reader, for instance, to [16].
In order to solve the stated problem by the Rothe method, we divide the interval I into n subintervals by points t j = jh n , j = 0, ..., n, where .., M and j = 1, ..., n.Introduce now functions obtained from the approximates u j by piecewise linear interpolation and piecewise EJQTDE, 2007 No. 14, p. 6 constant with respect to the time, respectively: and for j = 1, ..., n, and k = 1, ..., M.Moreover, we use the notation: t ∈ (t j−1 , t j ], k = 1, ..., M, thus for t ∈ (t j−1 , t j ], j = 1, ..., n. Finally, the following lemmas are used in this paper.We list them for convenience: Lemma 1 (An analogue of Gronwall's Lemma in continuous form [11]). Let f i (t) (i = 1, 2) be real continuous functions on the interval (0, T ) , f 3 (t) ≥ 0 nondecreasing function on t, and C > 0. Then the inequality Lemma 2 (Gronwall's Lemma in discret form [14]). Let {a i } be a sequence of real, nonnegative numbers, and A, C and h be positive constants.If the inequality takes place for all j = 1, 2, . . ., n, then the estimate holds for all j = 2, ... n.

Variational formulation
First, we take the scalar product in Applying the Green formula to the second term of the above identity, by taking into account condition (1.3b) and (2.6), we get Now, we are able to make precise the concept of the solution of problem (1.1)-(1.4)we are considering: (3.4) holds for all v ∈ V and a.e.t ∈ I.
Consider now the following linearized problem, obtained by discretization with respect to the time of (3.4) and consider the auxiliary functions then, we can easily get from which, it follows Therefore to prove the solvability of problem (3.5) it suffices to establish the proof for the following problem: Find, successively for j = 1, ..., n, the functions y j ∈ V verifying: EJQTDE, 2007 No. 14, p. 9 with In light of properties P1-P2, a successive application of the Lax-Milgram Theorem to the coupled problem (3.8)-(3.9)leads to: Lemma 3.Under properties P1-P2, problem (3.8)-(3.9)admits for all j = 1, ..., n, a unique solution As a consequence, we have Corollary 4. Problem (3.5) admits for all j = 1, ..., n, a unique solution u j ∈ H 2 (Ω) ∩ V.

A priori estimates for the discretized problem
Let us now derive some a priori estimates: Let assumption A4 and properties P1-P2 be fulfilled.Then, for n ∈ N * , the solutions u j of the semi-discretized problem (3.5), satisfy: for all j = 1, . . ., n, where C 1 is a positive constant independent of h n and j.
Proof.Take v = y j in the integral identity (3.9), it yields Thanks to the Schwarz inequality, we obtain Invoking (P2) and omitting the second term on the left-hand side of (4.2), it comes According to (3.9), we have Substituting (4.3) into (4.4),yields Iterating, we get According to assumption A4, the following inequality holds where Inserting (4.7) into (4.6), it comes Therefore, owing to Lemma 2.2, we obtain where , which concludes the proof.
Lemma 6.Under assumptions of Lemma 4.1, the following estimates ) where Therefore by virtue of (3.6), we have from where, due to (4.1) and (4.10), we deduce where On the other hand, invoking (4.2) we get, in consequence of the positivity of the first term and due to (3.9), it comes then, combining (4.12) and (4.13), and summing up the resulting inquality for i = 1, ..., j, to obtain consequently, by (4.1) and (4.7), we conclude with However, from (4.1), (4.7) and (4.12), it follows that from which, we have Finally, it follows from (3.6) that from which, due to (4.14) and (4.15), we find where This achieves the proof.

Convergence results
The variational equation (3.5) may be written in terms of u (n) and u (n) : For the functions u (n) and u (n) , we derive from results of Section 4 the following obvious properties: Corollary 7.For all n ∈ N * , the functions u (n) and u (n) satisfy the following estimates: ) where where As for estimate (5.4), it suffices to observe that therefore, due to (4.8), we obtain (5.4).Finally, it follows from Consequently, owing to (4.8), we get estimate (5.4).
To continue, we have need to establish the following lemma: Lemma 8. Let assumption A4 and property P1 be fulfilled.Then the following estimate holds for all v ∈ V and a.e.t ∈ I.
Proof.Identity (5.1) can be written In light of P1 and the Schwarz inequality, the right-hand side of (5.8) is then dominated as follows However due to (2.4) and (4.7), it yields for all j = 1, ..., n : therefore, owing to (4.1) Inserting (5.3) and (5.10) into (5.9),we obtain (5.7), with Let us subtract from identity (5.1) the similar identity for m, and Observing that EJQTDE, 2007 No. 14, p. 15 the last equality can be written for a.e.t ∈ I.
We now estimate the terms on the right-hand side of (5.11).In light of Lemma 5.
comes by taking into account (5.5): (5.12) where The first term on the right-hand side of (5.11) can be estimated as follows: But for all t ∈ I, there exist two integers j and i such that t ∈ (t j−1 , t j ]∩ (t i+1 , t i ] Consequently, inequality (5.13) becomes Substituting (5.12) and (5.14) into (5.11),integrating the result over (0, t) by taking into account the fact that u (n) (0) = u(0) = u 0 , and applying property P2, we obtain, by omitting the first term on the left hand-side and ,Ω ds, EJQTDE, 2007 No. 14, p. 17 for all t ∈ I. Hence, by virtue of Lemma 2.1, we get for all t ∈ I. Consequently, Since the right-hand side of the above inequality is independent of t; hence, replacing the left-hand side by its upper bound with respect to t from 0 to T, we obtain which implies that u (n) and u (n) are Cauchy sequences in the Banach spaces L 2 (I, V ) and C I, V , respectively.Consequently, having in mind (5.5), there exists some function u ∈ C I, V such that: as n tends to infinity.
According to (5.2b), (5.3) and (5.15), we get, by taking into account [14, Lemma 1.3.15], the following results formulated in: EJQTDE, 2007 No. 14, p. 18 Theorem 9. Let the assumption A4 et properties P1-P2 be hold.Then, the function u possesses the following properties: u is strongly differentiable a.e. in I and ) Proof.In light of (5.17) and (5.18) the points (i )-(ii ) of Definition 3.1 are verified.Furthermore, since by definition u (n) (0) = u 0 , it then follows from (5.15) that the point (iii ) of Definition 3.1 is fulfilled.It remains to prove that the limit function u = u(x, t) satisfies the integral identity (3.4).To this end, integrate identity (5.1) over (0, t) which can be written EJQTDE, 2007 No. 14, p. 19 We must show that It is easy to check Therefore invoking (5.15) and passing to the limit in (6.5), when n tends to infinity, we get However, owing to (5.16), property P1 and Lemma 5.2 we have Consequently, by combining (6.6) and (6.7), we obtain (6.3).On the other hand, owing to assumption A4, it comes EJQTDE, 2007 No. 14, p. 20 for all t ∈ (t j−1 , t j ] (j = 1, ..., n) .Consequently, according to (5.5) from which, we conclude, in view of (5.20), that in V, ∀t ∈ I.Moreover, by virtue of the Schwarz inequality and (5.10), we have Therefore the application of the Lebesgue Theorem of dominate convergence leads to (6.4).Hence, by passing to the limit in (6.2) when n tends to infinity, we get (u(t), v) 0,Ω + t 0 a (u(s), v) ds + αa (u(t), v) Differentiating the above identity with respect to t we obtain the integral identity (3.4), thanks to the identities s) 0,Ω ds, so that, owing to the Schwarz inequality and assumption A4 Omitting the first two terms on the left-hand side of the last inequality and using P2, we find Thanks to Lemma 2.1, we conclude that u(t) 1,Ω = 0, ∀t ∈ I, which implies the uniqueness of the solution.
Theorem 12. Let properties P1-P2 be fulfilled.Moreover, let u(x, t) and u * (x, t) be two solutions of problem (1.1)-(1.4)corresponding to (u 0 , f ) and (u * 0 , f * ), respectively.If there exists a continuous nonnegative function K(t) and a positive constant L such that the following estimate f (t, u, p) − f * (t, u * , p * ) 0,Ω (6.9) for all t ∈ I and takes pace for all u, u * , p and p * ∈ L 2 (Ω) , then where C 11 is a positive constant independent on u and u * .
Proof.Considering the variational formulation of problem (1.1)-(1.4)written for u, subtracting from it the same integral identity written for u * and setting v = u(t) − u * (t), we obtain after integrating the obtained identity over (0, t) Invoking properties P1-P2 and (6.9), we get, in consequence of the positivity of the first two terms on the left-hand side of (6.10) and of the application of the elementary inequalities 2ab ≤ a 2 + b 2 and (a + b + b) 2 ≤ 3 (a 2 + b 2 + c 2 ) to the right-hand side, after some rearrangement T , for all t ∈ I. Hence, in the left hand-side of (7.5), taking the upper bound with respect to t, we obtain

.20) 6 .
Existence, uniqueness and continuous dependence Theorem 10.Under assumptions of Theorem 5.3, the limit function u is the weak solution of problem (1.1)-(1.4) in the sense of Definition 3.1.