Electronic Journal of Qualitative Theory of Differential Equations 2007, No. 14, 1-27; http://www.math.u-szeged.hu/ejqtde/

SOLUTION TO A TRANSMISSION PROBLEM FOR QUASILINEAR PSEUDOPARABOLIC EQUATIONS BY

THE ROTHE METHOD

ABDELFATAH BOUZIANI AND NABIL MERAZGA

Abstract. 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 convergence of the method and an error estimate of the approximations are established.

1. 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 ∂Ω = \overline{Γ}^{0} ∪ \overline{Γ}^{1}, 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

N (Γ^{μ}) = {k, 1 ≤ k ≤ M/meas_{N−1} (Γ^{μ} ∩ ∂Ω_{k}) > 0} ,

Γ_{k}^{μ} = \{ \begin{matrix}Γ^{μ} ∩ ∂Ω_{k}, & k ∈ N (Γ^{μ}) , \\ ϕ, & k ∉ N (Γ^{μ}) , \end{matrix}

for μ = 0, 1. Moreover, let for k = 1, ..., M,

N_{k} = {ℓ, 1 ≤ ℓ ≤ M/ meas_{N−1} (∂Ω_{k} ∩ ∂Ω_{ℓ}) > 0}

Γ_{k,ℓ} = \{ \begin{matrix}∂Ω_{k} ∩ ∂Ω_{ℓ}, & ℓ ∈ N_{k}, \\ ϕ, & ℓ ∉ N_{k}. \end{matrix}

EJQTDE, 2007 No. 14, p. 1

2000 Mathematics Subject Classification. 35K70, 35A35, 35B30, 35B45, 35D05.

Key words and phrases. Quasilinear pseudoparabolic equation, transmission problem, Rothe method, a priori estimate, generalized solution.

fig. 1 For instance, in the situation illustrated by fig. 1, we have

N_{2} = {1, 2, 3}

and then:

Γ_{2,1} = ∂Ω_{2} ∩ ∂Ω_{1},

Γ_{2,2} = ∂Ω_{2},

Γ_{2,3} = ∂Ω_{2} ∩ ∂Ω_{3},
Γ_{2,ℓ} = ϕ, ∀ℓ = 4, ..., M.

One should note that ℓ ∈ N_{k} iff k ∈ N_{ℓ} and hence Γ_{k,ℓ} = Γ_{ℓ,k}, 1 ≤ k, ℓ ≤ 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

\frac{∂u^{k}}{∂t} + Au^{k} + αA\frac{∂u^{k}}{∂t} + a_{0}^{k}(x)u^{k} + αa_{0}^{k}(x)\frac{∂u^{k}}{∂t}

= f^{k}(x, t, u^{k}, ∇u^{k}) ,  in Ω_{k} × I,  (k = 1, ..., M) ,

where

Au^{k} = − ∑_{p,q=1}^{N} \frac{∂}{∂x_{p}} (a_{pq}^{k}(x)\frac{∂u^{k}}{∂x_{q}} ) ,

along with the initial conditions

u^{k}(x, 0) = u_{0}^{k}(x),  x ∈ Ω_{k}, (k = 1, ..., M) ,  (1.1)

EJQTDE, 2007 No. 14, p. 2 together with the Dirichlet and Neumann boundary conditions

\begin{matrix}u^{k} = 0,  on Γ_{k}^{0} × I (k = 1, ..., M) , \\ \frac{∂u^{k}}{∂ϑ_{A}} = 0,  on Γ_{k}^{1} × I (k = 1, ..., M) , \end{matrix}  (1.2)

as well as with transmission conditions

\begin{matrix}u^{k} = u^{ℓ},  on Γ_{k,ℓ} × I, ∀ℓ ∈ N_{k} (k = 1, ..., M) , \\ (\frac{∂u^{k}}{∂ϑ_{A}} + α\frac{∂^{2}u^{k}}{∂t∂ϑ_{A}} ) + (\frac{∂u^{ℓ}}{∂ϑ_{A}} + α\frac{∂^{2}u^{ℓ}}{∂t∂ϑ_{A}} ) = 0, \\ on Γ_{k,ℓ} × I, ∀ℓ ∈ N_{k} (k = 1, ..., M) , \end{matrix} (1.3)

where \frac{∂u^{k}}{∂ϑ_{A}} is the conormal derivative defined by:

\frac{∂u^{k}(x)}{∂ϑ_{A}} = ∑_{p,q=1}^{N} a_{pq}^{k}(x)\frac{∂u^{k}}{∂x_{q}} cos (ϑ^{k}, x_{p}) ,

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−αΔu 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

\frac{∂u}{∂t} − η \frac{∂^{3}u}{∂x^{2}∂t} − \frac{∂u}{∂x} + u\frac{∂u}{∂x} = 0  (1.4)

proposed in [3]. Taking into account dissipative phenomena, equation (1.5) is modified to the so-called Benjamin-Bona-Mahony-Burgers

EJQTDE, 2007 No. 14, p. 3 equation [1]

\frac{∂u}{∂t} − η \frac{∂^{3}u}{∂x^{2}∂t} − α\frac{∂^{2}u}{∂x^{2}} − \frac{∂u}{∂x} + u\frac{∂u}{∂x} = 0.  (1.5)

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.

2. 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:

V : = {v = (v^{1}, ..., v^{M})/ v^{k} ∈ V^{k}, v^{k}|_{Γ_{k,ℓ}} = v^{ℓ}|_{Γ_{k,ℓ}} ,  (2.1)
∀ℓ ∈ N_{k} (k = 1, ..., M)} ,

where

V^{k} := {v^{k} ∈ H^{1} (Ω_{k})/ v^{k} = 0 on Γ_{k}^{0}} .  (2.2)

The space V is equipped with the norm ∥·∥_{1,Ω} , namely

∥v∥_{1,Ω}^{2} = ∑_{k=1}^{M} ∥v^{k}∥_{1,Ω_{k}}^{2} .

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 𝕃^{2} (Ω) = L^{2} (Ω_{1}) × ⋯ × L^{2} (Ω_{M}) equipped with the scalar product and the associated norm

(u, v)_{0,Ω} = ∑_{k=1}^{M} (u^{k}, v^{k})_{0,Ω_{k}} and ∥u∥_{0,Ω}^{2} = ∑_{k=1}^{M} ∥u^{k}∥_{0,Ω_{k}}^{2} ,

respectively.

Now, we state the following hypotheses which are assumed to hold for k = 1, ..., M : A1. a_{pq}^{k} ∈ L^{∞} (Ω_{k}) ; ∃κ > 0, ∀ξ ∈ ℝ^{N} : ∑_{p,q=1}^{N} a_{pq}^{k}(x)ξ_{p}ξ_{q} ≥ κ ∑_{p=1}^{N} ξ_{p}^{2} a.e. in Ω_{k}, A2. a_{pq}^{k}(x) = a_{qp}^{k}(x), a.e. in Ω_{k}, A3. a_{0}^{k} ∈ L^{∞} (Ω_{k}) ; ∃β > 0 : a_{0}^{k}(x) ≥ β, a.e. in Ω_{k},

EJQTDE, 2007 No. 14, p. 5 A4. f^{k} (t, u^{k}, v^{k}) : I ×(L^{2} (Ω_{k}))^{2} → L^{2} (Ω_{k}) is bounded in L^{2} (Ω_{k}) and fulfills the Lipschitz condition:

∥f^{k} (t, u^{k}, v^{k}) − f^{k} (t′, u^{′k}, v^{′k})∥_{0,Ω_{k}}
≤ L (|t − t′| + ∥u^{k} − u^{′k}∥_{0,Ω_{k}} + ∥v^{k} − v^{′k}∥_{0,Ω_{k}})

for all t, t′ ∈ I, and u^{k}, u^{′k}, v^{k}, v^{′k} ∈ L^{2} (Ω_{k}) . A5. u_{0}^{k} ∈ V^{k}, k = 1, ..., M. Let us define the symmetrical integro-differential form

a (u, v) = ∑_{k=1}^{M} a_{k} (u^{k}, v^{k}) ,

where

a_{k} (u^{k}, v^{k}) = ∑_{p,q=1}^{N} ∫_{Ω_{k}} (a_{pq}^{k} \frac{∂u^{k}}{∂x_{q}} \frac{∂v^{k}}{∂x_{p}} + a_{0}^{k}u^{k}v^{k}) dx

with a_{pq}^{k} satisfy assumptions A1-A2, then the form a (u, v) fulfills the following properties: P1. ∀u, v ∈ V, |a (u, v)| ≤ κ_{0} ∥u∥_{1,Ω} ∥v∥_{1,Ω} , κ_{0} = cste, P2. There exists a sufficiently large constant β_{0} (≥ 1) such that

a (v, v) ≥ β_{0} ∥v∥_{1,Ω}^{2} , ∀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(\overline{I}, H), L^{∞}(I, H) and Lip(\overline{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 h_{n} := T/n is a time-step. Set

u_{j} = (u_{j}^{1}, u_{j}^{2}, ..., u_{j}^{M}) ,  u_{j}^{k} (x) :≃ u^{k} (x, t_{j}) ,
δu_{j} = (δu_{j}^{1}, δu_{j}^{2}, ..., δu_{j}^{M}) ,  δu_{j}^{k} (x) := \frac{u_{j}^{k} (x) − u_{j−1}^{k} (x)}{h_{n}},
f_{j} = (f_{j}^{1}, f_{j}^{2}, ..., f_{j}^{M}) ,  f_{j}^{k} (x) := f^{k} (x, t_{j}, u_{j−1}^{k}, ∇u_{j−1}^{k}) ,

for k = 1, ..., 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:

u^{(n)}(t) = (u^{1,n}(t), u^{2,n}(t), ⋯, u^{M,n}(t)) ,
\overline{u}^{(n)}(t) = (\overline{u}^{1,n}(t), \overline{u}^{2,n}(t), ⋯, \overline{u}^{M,n}(t)) ,

where

u^{k(n)}(t) := u_{j−1}^{k} + δu_{j}^{k} (t − t_{j−1}) , t ∈ [t_{j−1}, t_{j}] ,  (2.3)

and

\overline{u}^{k(n)}(t) := \{ \begin{matrix}u_{j}^{k}, & for t ∈ (t_{j−1}, t_{j}] , \\ u_{0}^{k}, & for t ∈ [−h_{n}, 0] , \end{matrix}  (2.4)

for j = 1, ..., n, and k = 1, ..., M. Moreover, we use the notation:

τ_{h_{n}}u^{k(n)} (x, t) = u^{k(n)} (x, t − h_{n}) , k = 1, ..., M,

then for w = (w^{1}, w^{2}, ⋯, w^{M}) ∈ ℍ^{1} (Ω) , we write

\overline{f}^{(n)}(t, w, ∇w) = (\overline{f}^{1(n)} (t, w^{1}, ∇w^{1}) , ⋯, \overline{f}^{M(n)} (t, w^{M}, ∇w^{M}))

with

\overline{f}^{k(n)}(t, w^{k}, ∇w^{k}) := f^{k}(t_{j}, w^{k}, ∇w^{k}),  (2.5)

t ∈ (t_{j−1}, t_{j}], k = 1, ..., M, thus

\overline{f}^{k(n)}(t, τ_{h_{n}} \overline{u}^{k(n)}, ∇τ_{h_{n}} \overline{u}^{k(n)}) := f^{k}(t_{j}, u_{j−1}^{k}, ∇u_{j−1}^{k}) = f_{j}^{k},  (2.6)

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

∫_{0}^{t} f_{1}(s)ds + f_{2}(t) ≤ f_{3}(t)e^{Ct}, ∀t ∈ (0, T) ,

is a consequence of the inequality

∫_{0}^{t} f_{1}(s)ds + f_{2}(t) ≤ f_{3}(t) + C ∫_{0}^{t} f_{2}(s)ds.

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

a_{j} ≤ A + Ch ∑_{i=1}^{j−1} a_{i},

takes place for all j = 1, 2, …, n, then the estimate

a_{i} ≤ Ae^{C(j−1)h},

holds for all j = 2, ... n.

3. Variational formulation

First, we take the scalar product in L^{2} (Ω_{k}) of equation (1.1) with v^{k} ∈ V^{k}, we have

(\frac{∂u^{k} (t)}{∂t}, v^{k})_{0,Ω_{k}} + (Au^{k}(t) + αA\frac{∂u^{k}(t)}{∂t}, v^{k})_{0,Ω_{k}} (3.1)

+ (a_{0}^{k} (u^{k} (t) + α\frac{∂u^{k} (t)}{∂t}) , v^{k})_{0,Ω_{k}}

= (f^{k}(t, u^{k} (t) , ∇u^{k} (t)), v^{k})_{0,Ω_{k}} .

Applying the Green formula to the second term of the above identity, by taking into account condition (1.3b) and (2.6), we get

(Au^{k}(t) + αA\frac{∂u^{k}(t)}{∂t}, v^{k})_{0,Ω_{k}}  (3.2)

= − ∑_{ℓ∈N_{k}} ∫_{Γ_{k,ℓ}} (\frac{∂u^{k} (t)}{∂ϑ_{A}} + α\frac{∂^{2}u^{k} (t)}{∂t∂ϑ_{A}}) v^{k}dx

+a_{k} (u^{k}(t), v^{k}) + αa_{k} (\frac{du^{k}(t)}{dt}, v^{k}) .

Substituting (3.2) into (3.1), it yields

(\frac{du^{k}(t)}{dt}, v^{k})_{0,Ω_{k}} + a_{k} (u^{k}(t), v^{k}) + αa_{k} (\frac{du^{k}(t)}{dt}, v^{k}) (3.3)
= (f^{k}(t, u^{k} (t) , ∇u^{k} (t)), v^{k})_{0,Ω_{k}} .

EJQTDE, 2007 No. 14, p. 8 Thus, summing up (3.3) for k = 1, ..., M and invoking (2.5) and (1.4), we obtain the desired variational formulation of problem (1.1)-(1.4)

(\frac{du(t)}{dt}, v)_{0,Ω} + a (u(t), v) + αa (\frac{du(t)}{dt}, v)  (3.4)
= (f(t, u (t) , ∇u (t)), v)_{0,Ω} , ∀v ∈ V, a.e.t ∈ I, u(t) ∈ V, t ∈ I.

Now, we are able to make precise the concept of the solution of problem (1.1)-(1.4) we are considering:

Definition 1. A function u : I → 𝕃^{2} (Ω) is called a weak solution of problem (1.1)-(1.4), if

(i) u ∈ Lip (\overline{I}, V ); (ii) u has a strong derivative (a.e. in I) \frac{du}{dt} ∈ L^{∞} (I, V) ; (iii) u(0) = (u_{0}^{1}, u_{0}^{2}, ..., u_{0}^{M}) = u_{0} in V; (iv) the identity (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)

\{ \begin{matrix}u_{j} ∈ V, \\ (δu_{j}, v)_{0,Ω} + a (u_{j}, v) + αa (δu_{j}, v) \\ = (f(t_{j}, u_{j−1}, ∇u_{j−1}), v)_{0,Ω} , ∀v ∈ V, (j = 1, ⋯, n), \end{matrix}  (3.5)

and consider the auxiliary functions

y_{j} = u_{j} + αδu_{j} (j = 1, ..., n) ,  (3.6)

then, we can easily get

u_{j} = \frac{h_{n}}{α + h_{n}} y_{j} + \frac{α}{α + h_{n}} u_{j−1} (j = 1, ..., n) ,

from which, it follows

δu_{j} = \frac{1}{α + h_{n}} (y_{j} − u_{j−1}) (j = 1, ..., n) .  (3.7)

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:

a (y_{j}, v) + \frac{1}{α + h_{n}} (y_{j}, v)_{0,Ω}  (3.8)
= (f_{j} + \frac{1}{α + h_{n}} u_{j−1}, v)_{0,Ω} ,  ∀v ∈ V,

with

u_{j} = \frac{h_{n}}{α + h_{n}} y_{j} + \frac{α}{α + h_{n}} u_{j−1}.  (3.9)

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 y_{j} ∈ H^{2} (Ω) ∩ V.

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.

4. A priori estimates for the discretized problem

Let us now derive some a priori estimates:

Lemma 5. 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:

∥u_{j}∥_{1,Ω} ≤ C_{1},  (4.1)

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

a (y_{j}, y_{j}) + \frac{1}{α + h_{n}} ∥y_{j}∥_{0,Ω}^{2} = (f_{j} + \frac{1}{α + h_{n}} u_{j−1}, y_{j})_{0,Ω} .

Thanks to the Schwarz inequality, we obtain

a (y_{j}, y_{j}) + \frac{1}{α + h_{n}} ∥y_{j}∥_{0,Ω}^{2}  (4.2)
≤ (∥f_{j}∥_{0,Ω} + \frac{1}{α + h_{n}} ∥u_{j−1}∥_{0,Ω}) ∥y_{j}∥_{0,Ω} .

Invoking (P2) and omitting the second term on the left-hand side of (4.2), it comes

αβ_{0} ∥y_{j}∥_{1,Ω} ≤ (α + h_{n}) β_{0} ∥y_{j}∥_{1,Ω} ≤ (α + h_{n}) ∥f_{j}∥_{0,Ω} + ∥u_{j−1}∥_{0,Ω} ,

therefore

∥y_{j}∥_{1,Ω} ≤ (α + h_{n}) ∥f_{j}∥_{0,Ω} + ∥u_{j−1}∥_{1,Ω} .  (4.3)

EJQTDE, 2007 No. 14, p. 10 According to (3.9), we have

∥u_{j}∥_{1,Ω} ≤ \frac{h_{n}}{α + h_{n}} ∥y_{j}∥_{1,Ω} + \frac{α}{α + h_{n}} ∥u_{j−1}∥_{1,Ω} (j = 1, ..., n) . (4.4)

Substituting (4.3) into (4.4), yields

∥u_{j}∥_{1,Ω} ≤ h_{n} ∥f_{j}∥_{0,Ω} + ∥u_{j−1}∥_{1,Ω} .  (4.5)

Iterating, we get

∥u_{j}∥_{1,Ω} ≤ h_{n} ∑_{i=1}^{j} ∥f_{i}∥_{0,Ω} + ∥u_{0}∥_{1,Ω} , ∀j = 1, ..., n.  (4.6)

According to assumption A4, the following inequality holds

∥f_{i}∥_{0,Ω} ≤ ∥f(t_{i}, u_{i−1}, ∇u_{i−1}) − f(t_{i}, 0, 0)∥_{0,Ω} + ∥f(t_{i}, 0, 0)∥_{0,Ω}
≤ \sqrt{2}L ∥u_{i−1}∥_{1,Ω} + M′,  (4.7)

where M′ := max_{t∈I} ∥f(t, 0, 0)∥_{0,Ω} < ∞.

Inserting (4.7) into (4.6), it comes

∥u_{j}∥_{1,Ω} ≤ h_{n} ∑_{i=1}^{j} (\sqrt{2}L ∥u_{i−1}∥_{1,Ω} + M′) + ∥u_{0}∥_{1,Ω}
= M′jh_{n} + ∥u_{0}∥_{1,Ω} + \sqrt{2}h_{n}L ∑_{i=0}^{j−1} ∥u_{i}∥_{1,Ω}
≤ M′T + (1 + \sqrt{2}LT ) ∥u_{0}∥_{1,Ω} + \sqrt{2}h_{n}L ∑_{i=1}^{j−1} ∥u_{i}∥_{1,Ω} .

Therefore, owing to Lemma 2.2, we obtain ∥u_{j}∥_{1,Ω} ≤ (M′T + (1 + \sqrt{2}LT ) ∥u_{0}∥_{1,Ω}) e^{\sqrt{2}L(j−1)h_{n}} ≤ C_{1},

where C_{1} := (M′T + (1 + \sqrt{2}LT ) ∥u_{0}∥_{1,Ω}) e^{\sqrt{2}LT} , which concludes the proof. Lemma 6. Under assumptions of Lemma 4.1, the following estimates

∥δu_{j}∥_{1,Ω} ≤ C_{2},  (4.8)
∥δu_{j}∥_{0,Ω} ≤ C_{3},  (4.9)

hold for j = 1, …, n, where C_{2} and C_{3} are positive constants independent of h_{n} and j.

EJQTDE, 2007 No. 14, p. 11 Proof. According to (4.1) and (4.7), inequality (4.3) becomes

∥y_{j}∥_{1,Ω} ≤ (α + h_{n}) (\sqrt{2}L ∥u_{j−1}∥_{1,Ω} + M′) + ∥u_{j−1}∥_{1,Ω} ≤ C_{4},

where

C_{4} := ((α + T) \sqrt{2}L + 1) C_{1} + (α + T) M′.

Therefore by virtue of (3.6), we have

∥δu_{j}∥_{1,Ω} ≤ \frac{1}{α} (∥y_{j}∥_{1,Ω} + ∥u_{j−1}∥_{1,Ω}) ,

from where, due to (4.1) and (4.10), we deduce

∥δu_{j}∥_{1,Ω} ≤ C_{2}, ∀j = 1, ..., n,  (4.10)

where

C_{2} := \frac{1}{α} (C_{1} + C_{4}) .

On the other hand, invoking (4.2) we get, in consequence of the positivity of the first term

∥y_{j}∥_{1,Ω} ≤ (α + h_{n}) ∥f_{j}∥_{0,Ω} + ∥u_{j−1}∥_{0,Ω}  (4.11)

and due to (3.9), it comes

∥u_{j}∥_{0,Ω} ≤ \frac{h_{n}}{α + h_{n}} ∥u_{j}∥_{0,Ω} + \frac{h_{n}}{α + h_{n}} ∥u_{j−1}∥_{0,Ω} (j = 1, ..., n) , (4.12)

then, combining (4.12) and (4.13), and summing up the resulting inquality for i = 1, ..., j, to obtain

∥u_{j}∥_{0,Ω} ≤ h_{n} ∑_{i=1}^{j} ∥f_{ι}∥_{0,Ω} + ∥u_{0}∥_{0,Ω}

consequently, by (4.1) and (4.7), we conclude

∥u_{j}∥_{0,Ω} ≤ C_{5,}  ∀j = 1, .., n,  (4.13)

with

C_{5} := \sqrt{2}LC_{1}T + M′T + ∥u_{0}∥_{0,Ω} .

However, from (4.1), (4.7) and (4.12), it follows that

∥y_{j}∥_{0,Ω} ≤ (α + h_{n}) (\sqrt{2}L ∥u_{j−1}∥_{1,Ω} + M′) + ∥u_{j−1}∥_{0,Ω} ,

from which, we have

∥y_{j}∥_{0,Ω} ≤ C_{6,}  ∀j = 1, .., n,  (4.14)

EJQTDE, 2007 No. 14, p. 12 with

C_{6} := (α + T) (\sqrt{2}LC_{1} + M′) + C_{5}.

Finally, it follows from (3.6) that

∥δu_{j}∥_{0,Ω} ≤ \frac{1}{α} (∥y_{j}∥_{0,Ω} + ∥u_{j}∥_{0,Ω})

from which, due to (4.14) and (4.15), we find

∥δu_{j}∥_{0,Ω} ≤ C_{3,} ∀j = 1, .., n,

where

C_{3} := \frac{1}{α} (C_{5} + C_{6}) .

This achieves the proof.

5. Convergence results

The variational equation (3.5) may be written in terms of u^{(n)} and \overline{u}^{(n)} :

(\frac{du^{(n)}(t)}{dt}, v)_{0,Ω} + a (\overline{u}^{(n)}(t), v) + αa (\frac{du^{(n)}(t)}{dt}, v)  (5.1)
= (\overline{f}^{(n)} (t, τ_{h_{n}} \overline{u}^{(n)}(t), ∇τ_{h_{n}} \overline{u}^{(n)}(t)) , v)_{0,Ω} ,  ∀v ∈ V, ∀t ∈ I.

For the functions u^{(n)} and \overline{u}^{(n)}, we derive from results of Section 4 the following obvious properties:

Corollary 7. For all n ∈ ℕ^{∗}, the functions u^{(n)} and \overline{u}^{(n)} satisfy the following estimates:

∥u^{(n)}(t)∥_{1,Ω} ≤ C_{1},  ∥\overline{u}^{(n)}(t)∥_{1,Ω} ≤ C_{1},  ∀t ∈ I,  (5.2)

∥\frac{du^{(n)}(t)}{dt}∥_{1,Ω} ≤ C_{2},  ∥\frac{du^{(n)}(t)}{dt}∥_{0,Ω} ≤ C_{3},  a.e. in I, (5.3)
∥\overline{u}^{(n)}(t) − u^{(n)}(t)∥_{1,Ω} ≤ C_{2}h_{n},  ∀t ∈ I,  (5.4)
∥\overline{u}^{(n)}(t) − τ_{h_{n}} \overline{u}^{(n)}(t)∥_{1,Ω} ≤ C_{2}h_{n},  ∀t ∈ I,  (5.5)

where C_{1}, C_{2} and C_{3} are the same constants given in Lemmas 4.1 and 4.2.

EJQTDE, 2007 No. 14, p. 13 Proof. Estimates (5.2) follow directly from (4.1). To establish estimates (5.3), we differentiate identity (2.1) with respect to t and take into account (4.8) and (4.9), it yields

∥\frac{du^{(n)}(t)}{dt}∥_{1,Ω} ≤ C_{2},  ∥\frac{du^{(n)}(t)}{dt}∥_{0,Ω} ≤ C_{3}, for a.e. t ∈ I,

from where we get

∫_{I} ∥\frac{du^{(n)}(t)}{dt}∥_{1,Ω}^{2} dt ≤ C_{8},  ∫_{I} ∥\frac{du^{(n)}(t)}{dt}∥_{0,Ω}^{2} dt ≤ C_{7},  (5.6)

where C_{7} = C_{3}^{2}T and C_{8} = C_{2}^{2}T. As for estimate (5.4), it suffices to observe that

\overline{u}^{(n)}(t) − u^{(n)}(t) = (t_{j} − t)δu_{j}, ∀t ∈ (t_{j−1}, t_{j}] (j = 1, ..., n) ,

so that

∥\overline{u}^{(n)}(t) − u^{(n)}(t)∥ ≤ h_{n} max_{1≤j≤n} ∥δu_{j}∥_{1,Ω} ∀ t ∈ I (j = 1, ..., n) ,

therefore, due to (4.8), we obtain (5.4). Finally, it follows from

\overline{u}^{(n)}(t) − τ_{h_{n}} \overline{u}^{(n)} (t) = u_{j} − u_{j−1}, ∀t ∈ (t_{j−1}, t_{j}] (j = 1, ..., n) ,

that

∥\overline{u}^{(n)}(t) − τ_{h_{n}} \overline{u}^{(n)} (t)∥_{1,Ω} ≤ h_{n} max_{1≤j≤n} ∥δu_{j}∥_{1,Ω} , ∀ t ∈ I.

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

|a (\overline{u}^{(n)}(t), v)| ≤ C_{9} ∥v∥_{1,Ω} ,  (5.7)

holds for all v ∈ V and a.e. t ∈ I.

Proof. Identity (5.1) can be written

a (\overline{u}^{(n)}(t), v) = (\overline{f}^{(n)} (t, τ_{h_{n}} \overline{u}^{(n)}(t), ∇τ_{h_{n}} \overline{u}^{(n)}(t)) − \frac{du^{(n)}(t)}{dt}, v)_{0,Ω}

−αa (\frac{du^{(n)}(t)}{dt}, v) ,  ∀t ∈ I, ∀v ∈ V.  (5.8)

EJQTDE, 2007 No. 14, p. 14 In light of P1 and the Schwarz inequality, the right-hand side of (5.8) is then dominated as follows

|a (\overline{u}^{(n)}(t), v)| ≤ (∥\overline{f}^{(n)} (t, τ_{h_{n}} \overline{u}^{(n)}(t), ∇τ_{h_{n}} \overline{u}^{(n)}(t))∥_{0,Ω} (5.9)

+ ∥\frac{du^{(n)}(t)}{dt}∥_{0,Ω} + ακ_{0} ∥\frac{du^{(n)}(t)}{dt}∥_{1,Ω}) ∥v∥_{1,Ω} .

However due to (2.4) and (4.7), it yields for all j = 1, ..., n :

∥\overline{f}^{(n)} (t, τ_{h_{n}} \overline{u}^{(n)} (t) , ∇τ_{h_{n}} \overline{u}^{(n)} (t))∥_{0,Ω} = ∥f_{j}∥_{0,Ω}
≤ \sqrt{2}L ∥u_{j−1}∥_{1,Ω} + M′, ∀t ∈ (t_{j−1}, t_{j}] ;

therefore, owing to (4.1)

∥\overline{f}^{(n)} (t, τ_{h_{n}} \overline{u}^{(n)} (t) , ∇τ_{h_{n}} \overline{u}^{(n)} (t))∥_{0,Ω} ≤ \sqrt{2}LC_{1} +M′, ∀t ∈ I. (5.10)

Inserting (5.3) and (5.10) into (5.9), we obtain (5.7), with

C_{9} := (\sqrt{2}LC_{1} + M′ + C_{3} + ακ_{0}C_{2}) .

Let us subtract from identity (5.1) the similar identity for m, and set v = u^{(n)}(t) − u^{(m)}(t) (∈ V) , we have for all t ∈ I

(\frac{d}{dt} (u^{(n)}(t) − u^{(m)}(t)) , u^{(n)}(t) − u^{(m)}(t))_{0,Ω}

+a (\overline{u}^{(n)}(t) − \overline{u}^{(m)}(t), u^{(n)}(t) − u^{(m)}(t))

+αa (\frac{d}{dt} (u^{(n)}(t) − u^{(m)}(t)) , u^{(n)}(t) − u^{(m)}(t))

= (\overline{f}^{(n)} (t, τ_{h_{n}} \overline{u}^{(n)}(t), ∇τ_{h_{n}} \overline{u}^{(n)}(t))

− \overline{f}^{(m)} (t, τ_{h_{m}} \overline{u}^{(m)}(t), ∇τ_{h_{m}} \overline{u}^{(m)}(t)) , u^{(n)}(t) − u^{(m)}(t))_{0,Ω} .

Observing that

u^{(n)} − u^{(m)} = (\overline{u}^{(n)} − \overline{u}^{(m)}) − (\overline{u}^{(n)} − u^{(n)}) − (u^{(m)} − \overline{u}^{(m)}) ,

EJQTDE, 2007 No. 14, p. 15 the last equality can be written

\frac{d}{dt} ∥u^{(n)}(t) − u^{(m)}(t)∥_{0,Ω}^{2}  (5.11)

+2a (\overline{u}^{(n)}(t) − \overline{u}^{(m)}(t), \overline{u}^{(n)}(t) − \overline{u}^{(m)}(t))

+α\frac{d}{dt}a (u^{(n)}(t) − u^{(m)}(t), u^{(n)}(t) − u^{(m)}(t))

= 2 (\overline{f}^{(n)} (t, τ_{h_{n}} \overline{u}^{(n)} (t) , ∇τ_{h_{n}} \overline{u}^{(n)} (t))
− \overline{f}^{(m)} (t, τ_{h_{m}} \overline{u}^{(m)} (t) , ∇τ_{h_{m}} \overline{u}^{(m)} (t)) , u^{(n)}(t) − u^{(m)}(t))_{0,Ω}

+2a (\overline{u}^{(n)}(t) − \overline{u}^{(m)}(t), (\overline{u}^{(n)} (t) − u^{(n)} (t)) + (u^{(m)} (t) − \overline{u}^{(m)} (t))) ,

for a.e. t ∈ I.

We now estimate the terms on the right-hand side of (5.11). In light of Lemma 5.2 with v = (\overline{u}^{(n)} (t) − u^{(n)} (t)) + (u^{(m)} (t) − \overline{u}^{(m)} (t)), it comes by taking into account (5.5):

|a (\overline{u}^{(n)}(t) − \overline{u}^{(m)} (t) , (\overline{u}^{(n)} (t) − u^{(n)} (t)) + (u^{(m)} (t) − \overline{u}^{(m)} (t)))|

≤ 2C_{9} (∥\overline{u}^{(n)}(t) − u^{(n)}(t)∥_{1,Ω} + ∥u^{(m)} (t) − \overline{u}^{(m)} (t)∥_{1,Ω})  (5.12)
≤ C_{10} (h_{n} + h_{m}) ,

where C_{10} := 2C_{2}C_{9}. The first term on the right-hand side of (5.11) can be estimated as follows:

2 (\overline{f}^{(n)} (t, τ_{h_{n}} \overline{u}^{(n)} (t) , ∇τ_{h_{n}} \overline{u}^{(n)} (t))  (5.13)
−\overline{f}^{(m)} (t, τ_{h_{m}} \overline{u}^{(m)} (t) , ∇τ_{h_{m}} \overline{u}^{(m)} (t)) , u^{(n)}(t) − u^{(m)}(t))_{0,Ω}
≤ ∥\overline{f}^{(n)} (t, τ_{h_{n}} \overline{u}^{(n)} (t) , ∇τ_{h_{n}} \overline{u}^{(n)} (t))
−\overline{f}^{(m)} (t, τ_{h_{m}} \overline{u}^{(m)} (t) , ∇τ_{h_{m}} \overline{u}^{(m)} (t))∥_{0,Ω}^{2}
+ ∥u^{(n)}(t) − u^{(m)}(t)∥_{1,Ω}^{2} .

EJQTDE, 2007 No. 14, p. 16 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}]

∥\overline{f}^{(n)} (t, τ_{h_{n}} \overline{u}^{(n)} (t) , ∇τ_{h_{n}} \overline{u}^{(n)} (t)) − \overline{f}^{(m)} (t, τ_{h_{m}} \overline{u}^{(m)} (t) , ∇τ_{h_{m}} \overline{u}^{(m)} (t))∥_{0,Ω}
= ∥f (t_{j}, τ_{h_{n}} \overline{u}^{(n)} (t) , ∇τ_{h_{n}} \overline{u}^{(n)} (t)) − \overline{f} (t_{k}, τ_{h_{m}} \overline{u}^{(m)} (t) , ∇τ_{h_{m}} \overline{u}^{(m)} (t))∥_{0,Ω}
≤ \sqrt{3}L (\sqrt{M′} (h_{n} + h_{m}) + ∥τ_{h_{n}} \overline{u}^{(n)} (t) − τ_{h_{m}} \overline{u}^{(m)} (t)∥_{1,Ω})
≤ \sqrt{3}L (\sqrt{M′} (h_{n} + h_{m}) + ∥τ_{h_{n}} \overline{u}^{(n)} (t) − \overline{u}^{(n)} (t)∥_{1,Ω}
+ ∥\overline{u}^{(n)} (t) − u^{(n)} (t)∥_{1,Ω} + ∥u^{(n)} (t) − u^{(m)} (t)∥_{1,Ω}
+ ∥u^{(m)} (t) − \overline{u}^{(m)} (t)∥_{1,Ω} + ∥\overline{u}^{(m)} (t) − τ_{h_{m}} \overline{u}^{(m)} (t)∥_{1,Ω})
≤ \sqrt{3}L ((\sqrt{M′} + 2C_{2}) (h_{n} + h_{m}) + ∥u^{(n)} (t) − u^{(m)} (t)∥_{1,Ω}) .

Consequently, inequality (5.13) becomes

2 (\overline{f}^{(n)} (t, τ_{h_{n}} \overline{u}^{(n)} (t) , ∇τ_{h_{n}} \overline{u}^{(n)} (t))  (5.14)
−\overline{f}^{(m)} (t, τ_{h_{m}} \overline{u}^{(m)} (t) , ∇τ_{h_{m}} \overline{u}^{(m)} (t)) , u^{(n)}(t) − u^{(m)}(t))_{0,Ω}
≤ 6L^{2} (\sqrt{M′} + 2C_{2})^{2} (h_{n} + h_{m})^{2}
+ (6L^{2} + 1) ∥u^{(n)} (t) − u^{(m)} (t)∥_{1,Ω}^{2} .

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

∫_{0}^{t} ∥\overline{u}^{(n)}(s) − \overline{u}^{(m)}(s)∥_{1,Ω}^{2} ds + ∥u^{(n)}(t) − u^{(m)}(t)∥_{1,Ω}^{2}

≤ \frac{2T}{β_{0} min (2, α)} (C_{10} (h_{n} + h_{m}) + 3L^{2} (\sqrt{M′} + 2C_{2})^{2} (h_{n} + h_{m})^{2})

+\frac{(6L^{2} + 1)}{β_{0} min (2, α)} ∫_{0}^{t} ∥u^{(n)}(s) − u(s)∥_{1,Ω}^{2} ds,

EJQTDE, 2007 No. 14, p. 17 for all t ∈ I. Hence, by virtue of Lemma 2.1, we get

∫_{0}^{t} ∥\overline{u}^{(n)}(s) − \overline{u}^{(m)}(s)∥_{1,Ω}^{2} ds + ∥u^{(n)}(t) − u^{(m)}(t)∥_{1,Ω}^{2}

≤ \frac{2T}{β_{0} min (2, α)} (C_{10} (h_{n} + h_{m})

+3L^{2} (\sqrt{M′} + 2C_{2})^{2} (h_{n} + h_{m})^{2}) exp (\frac{(6L^{2} + 1) t}{β_{0} min (2, α)}) ,

for all t ∈ I. Consequently,

∫_{0}^{t} ∥\overline{u}^{(n)}(s) − \overline{u}^{(m)}(s)∥_{1,Ω}^{2} ds + ∥u^{(n)}(t) − u^{(m)}(t)∥_{1,Ω}^{2}

≤ \frac{2T}{β_{0} min (2, α)} (C_{10} (h_{n} + h_{m})

+3L^{2} (\sqrt{M′} + 2C_{2})^{2} (h_{n} + h_{m})^{2}) exp (\frac{(2L^{2} + 1)}{β_{0} min (2, α)}T ) .

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

∥\overline{u}^{(n)} − \overline{u}^{(m)}∥_{L^{2}(I,V)}^{2} + ∥u^{(n)} − u^{(m)}∥_{C(\overline{I},V)}^{2}

≤ \frac{2T}{β_{0} min (2, α)} (C_{10} (h_{n} + h_{m})

+3L^{2} (\sqrt{M′} + 2C_{2})^{2} (h_{n} + h_{m})^{2}) exp (\frac{(2L^{2} + 1) T}{αβ_{0}} ) ,

which implies that \{\overline{u}^{(n)}\} and \{u^{(n)}\} are Cauchy sequences in the Banach spaces L^{2} (I, V) and C (\overline{I}, V ) , respectively. Consequently, having in mind (5.5), there exists some function u ∈ C (\overline{I}, V ) such that:

u^{(n)} → u  in C (\overline{I}, V )  (5.15)
\overline{u}^{(n)} → u  in L^{2} (I, V)  (5.16)

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 ∈ Lip (\overline{I}, V ) ,  (5.17)
u is strongly differentiable a.e. in I and \frac{du}{dt} ∈ L^{∞} (I, V) , (5.18)
\overline{u}^{(n)}(t) → u(t),  in V, ∀t ∈ I,  (5.19)
\frac{du^{(n)}}{dt} ⇀ \frac{du}{dt},  in L^{2} (I, V) .  (5.20)

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.

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) ⊂ I

(u^{(n)}(t), v)_{0,Ω} + ∫_{0}^{t} a (\overline{u}^{(n)}(s), v) ds + αa (u^{(n)}(t), v) (6.1)
= ∫_{0}^{t} (\overline{f}^{(n)} (s, τ_{h_{n}} \overline{u}^{(n)} (s) , ∇τ_{h_{n}} \overline{u}^{(n)} (s)) , v)_{0,Ω} ds
+ (u_{0}, v)_{0,Ω} + αa (u_{0}, v) ,

which can be written

(u^{(n)}(t) − u(t), v)_{0,Ω} + (u(t), v)_{0,Ω}  (6.2)
+ ∫_{0}^{t} a (\overline{u}^{(n)}(s) − u(s), v) ds + ∫_{0}^{t} a (u(s), v) ds
+αa (u^{(n)}(t) − u(t), v) + αa (u(t), v)
= ∫_{0}^{t} (\overline{f}^{(n)} (s, τ_{h_{n}} \overline{u}^{(n)} (s) , ∇τ_{h_{n}} \overline{u}^{(n)} (s))
− f (s, u (s) , ∇u (s)) , v)_{0,Ω} ds

+ ∫_{0}^{t} (f (s, u (s) , ∇u (s)) , v)_{0,Ω} ds + (u_{0}, v)_{0,Ω} + αa (u_{0}, v) .

EJQTDE, 2007 No. 14, p. 19 We must show that

lim_{n→∞} ((u^{(n)}(t) − u(t), v)_{0,Ω} + ∫_{0}^{t} a (u^{(n)}(s) − u(s), v) ds (6.3)
+ αa (u^{(n)}(t) − u(t), v) = 0

and

lim_{n→∞} ∫_{0}^{t} (\overline{f}^{(n)} (s, τ_{h_{n}} \overline{u}^{(n)} (s) , ∇τ_{h_{n}} \overline{u}^{(n)} (s))  (6.4)
− f (t, u (s) , ∇u (s)) , v)_{0,Ω} ds = 0.

It is easy to check

(u^{(n)}(t) − u(t), v)_{0,Ω} + αa (u^{(n)}(t) − u(t), v)  (6.5)
≤ (1 + ακ_{0}) ∥u^{(n)} (t) − u(t)∥_{1,Ω} ∥v∥_{1,Ω} , ∀t ∈ I.

Therefore invoking (5.15) and passing to the limit in (6.5), when n tends to infinity, we get

lim_{n→∞} ((u^{(n)}(t) − u(t), v)_{0,Ω} + αa (u^{(n)}(t) − u(t), v)) = 0.  (6.6)

However, owing to (5.16), property P1 and Lemma 5.2 we have

|∫_{0}^{t} a (\overline{u}^{(n)}(s) − u(s), v) ds| ≤ κ_{0}\sqrt{T} ∥v∥_{1,Ω} ∥\overline{u}^{(n)} − u∥_{L^{2}(I,V)} ,

hence

lim_{n→∞} ∫_{0}^{t} a (\overline{u}^{(n)}(s) − u(s), v) ds = 0, ∀v ∈ V, ∀t ∈ I.  (6.7)

Consequently, by combining (6.6) and (6.7), we obtain (6.3). On the other hand, owing to assumption A4, it comes

∥\overline{f}^{(n)} (t, τ_{h_{n}} \overline{u}^{(n)} (t) , ∇τ_{h_{n}} \overline{u}^{(n)} (t)) − f (t, u (t) , ∇u (t))∥_{0,Ω}
= ∥f (t_{j}, τ_{h_{n}} \overline{u}^{(n)} (t) , ∇τ_{h_{n}} \overline{u}^{(n)} (t)) − f (t, u (t) , ∇u (t))∥_{0,Ω}
≤ \sqrt{3}L (\sqrt{M′} |t_{j} − t| + ∥τ_{h_{n}} \overline{u}^{(n)} (t) − u(t)∥_{1,Ω}) ,

EJQTDE, 2007 No. 14, p. 20 for all t ∈ (t_{j−1}, t_{j}] (j = 1, ..., n) . Consequently, according to (5.5)

∥\overline{f}^{(n)} (t, τ_{h_{n}} \overline{u}^{(n)} (t) , ∇τ_{h_{n}} \overline{u}^{(n)} (t)) − f (t, u (t) , ∇u (t))∥_{0,Ω}
≤ \sqrt{3}L (\sqrt{M′}h_{n} + ∥τ_{h_{n}} \overline{u}^{(n)} (t) − \overline{u}^{(n)}(t)∥_{1,Ω} + ∥\overline{u}^{(n)} (t) − u(t)∥_{1,Ω})
≤ \sqrt{3}Lh_{n} (\sqrt{M′} + C_{2}) + \sqrt{3}L ∥\overline{u}^{(n)} (t) − u(t)∥_{1,Ω} ,

from which, we conclude, in view of (5.20), that

\overline{f}^{(n)} (t, τ_{h_{n}} \overline{u}^{(n)} (t) , ∇τ_{h_{n}} \overline{u}^{(n)} (t)) ⟶_{n→∞} f (t, u (t) , ∇u (t)) ,  (6.8)

in V, ∀t ∈ I. Moreover, by virtue of the Schwarz inequality and (5.10), we have

(\overline{f}^{(n)} (t, τ_{h_{n}} \overline{u}^{(n)} (t) , ∇τ_{h_{n}} \overline{u}^{(n)} (t)) , v)_{0,Ω}
≤ (\sqrt{2}LC_{1} + M′) ∥v∥_{0,Ω} ,  ∀t ∈ I, ∀n.

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} + ∫_{0}^{t} a (u(s), v) ds + αa (u(t), v)
= ∫_{0}^{t} (f (s, u (s) , ∇u (s)) , v)_{0,Ω} ds + (u_{0}, v)_{0,Ω} + αa (u_{0}, v) .

Differentiating the above identity with respect to t we obtain the integral identity (3.4), thanks to the identities

\frac{d}{dt} (u(t), v)_{0,Ω} = (\frac{du(t)}{dt}, v)_{0,Ω} , ∀v ∈ V, a.e. t ∈ I,

and

\frac{d}{dt}a (u(t), v) = a (\frac{du(t)}{dt}, v) , ∀v ∈ V, a.e. t ∈ I.

This completes the proof of Theorem 6.1.

Theorem 11. Under assumptions of Theorem 5.3, the weak solution of problem (1.1)-(1.4) is unique.

EJQTDE, 2007 No. 14, p. 21 Proof. Assume that (1.1)-(1.4) possesses two weak solutions u^{1} and u^{2}. Taking the difference of identities (3.4) corresponding to u^{1} and u^{2}, and v = u, where u = u^{1} − u^{2}, it yields

\frac{d}{dt} ∥u(t)∥_{0,Ω}^{2} + 2a (u(t), u(t)) + α\frac{d}{dt}a (u(t), u(t))

= 2 (f (t, u^{1} (t) , ∇u^{1} (t)) − f (t, u^{2} (t) , ∇u^{2} (t)) , u (t))_{0,Ω} ,

a.e. t ∈ I. Integrating over (0, t) ⊂ I, we get by taking into account that u(0) = 0

∥u(t)∥_{0,Ω}^{2} + 2 ∫_{0}^{t} a (u(s), u(s)) ds + αa (u(t), u(t))
= 2 ∫_{0}^{t} (f (s, u^{1} (s) , ∇u^{1} (s)) − f (s, u^{2} (s) , ∇u^{2} (s)) , u (s))_{0,Ω} ds,

so that, owing to the Schwarz inequality and assumption A4

∥u(t)∥_{0,Ω}^{2} + 2 ∫_{0}^{t} a (u(s), u(s)) ds + αa (u(t), u(t))
≤ 2 ∫_{0}^{t} ∥f (s, u^{1} (s) , ∇u^{1} (s)) − f (s, u^{2} (s) , ∇u^{2} (s))∥_{0,Ω} ∥u (s)∥_{0,Ω} ds
≤ 2\sqrt{2}L ∫_{0}^{t} ∥u(s)∥_{1,Ω}^{2} ds, ∀t ∈ I.

Omitting the first two terms on the left-hand side of the last inequality and using P2, we find

∥u(t)∥_{1,Ω}^{2} ≤ \frac{2\sqrt{2}L}{αβ_{0}} ∫_{0}^{t} ∥u(s)∥_{1,Ω}^{2} ds, ∀t ∈ I.

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)
≤ K(t) + L′ (∥u − u^{∗}∥_{0,Ω} + ∥p − p^{∗}∥_{0,Ω})

EJQTDE, 2007 No. 14, p. 22 for all t ∈ I and takes pace for all u, u^{∗}, p and p^{∗} ∈ 𝕃^{2} (Ω) , then

∥u − u^{∗}∥_{C(\overline{I},V)}^{2} ≤ C_{11} (∫_{0}^{t} K^{2}(s)ds + ∥u_{0} − u_{0}^{∗}∥_{1,Ω}^{2}) ,  (6.10)

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) ⊂ I

∥u(t) − u^{∗} (t)∥_{0,Ω}^{2} + 2 ∫_{0}^{t} a (u(s) − u^{∗} (s) , u(s) − u^{∗} (s)) ds

+αa (u(t) − u^{∗} (t) , u(t) − u^{∗} (t))  (6.11)
= 2 ∫_{0}^{t} (f (s, u (s) , ∇u (s))
−f^{∗} (s, u^{∗} (s) , ∇u^{∗} (s)) , u (s) − u^{∗} (s))_{0,Ω} ds
+ ∥u_{0} − u_{0}^{∗}∥_{0,Ω}^{2} + αa (u_{0} − u_{0}^{∗}, u_{0} − u_{0}^{∗}) .

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

∥u(t) − u^{∗}(t)∥_{1,Ω}^{2}
≤ max (\frac{3}{αβ_{0}} , \frac{1 + ακ_{0}}{αβ_{0}} ) (∫_{0}^{t} K^{2}(s)ds + ∥u_{0} − u_{0}^{∗}∥_{1,Ω}^{2})

+\frac{3L^{′2} + 1}{αβ_{0}} ∫_{0}^{t} ∥u(s) − u^{∗}(s)∥_{1,Ω}^{2} ds.

Thanks to Lemma 2.1 we get inequality (6.10), with

C_{11} := max (\frac{3}{αβ_{0}} , \frac{1 + ακ_{0}}{αβ_{0}} ) exp (\frac{(3L^{′2} + 1) T}{αβ_{0}} ) .

7. Error estimate

Theorem 13. Under the assumption A4 and properties P1-P2, the error of the approximation

∥u^{(n)} − u∥_{C(\overline{I},V)} ≤ C_{12}h_{n}  (7.1)

is valid for all n ∈ ℕ^{∗}.

Proof. Considering the difference between (5.1) and (3.4), and putting v = u^{(n)}(t) − u(t) (∈ V) , we have

(\frac{d}{dt} (u^{(n)}(t) − u(t)) , u^{(n)}(t) − u(t))_{0,Ω}

+a (\overline{u}^{(n)}(t) − u(t), u^{(n)}(t) − u(t))

+αa (\frac{d}{dt} (u^{(n)}(t) − u(t)) , u^{(n)}(t) − u(t))

= (\overline{f}^{(n)} (t, τ_{h_{n}} \overline{u}^{(n)}(t), ∇τ_{h_{n}} \overline{u}^{(n)}(t)) − f (t, u (t) , ∇u (t)) , u^{(n)}(t) − u(t))_{0,Ω} ,

from which we obtain

\frac{1}{2} \frac{d}{dt} ∥u^{(n)}(t) − u(t)∥_{0,Ω}^{2} + a (u^{(n)}(t) − u(t), u^{(n)}(t) − u(t))

+αa (\frac{d}{dt} (u^{(n)}(t) − u(t)) , u^{(n)}(t) − u(t))

= (\overline{f}^{(n)} (t, τ_{h_{n}} \overline{u}^{(n)}(t), ∇τ_{h_{n}} \overline{u}^{(n)}(t)) − f (t, u (t) , ∇u (t)) , u^{(n)}(t) − u(t))_{0,Ω}

+a (u^{(n)}(t) − \overline{u}^{(n)}(t), u^{(n)}(t) − u(t)) , a.e. t ∈ I.

Owing to properties P1-P2 and the Schwarz and Cauchy inequalities, it yields

\frac{1}{2} \frac{d}{dt} ∥u^{(n)}(t) − u(t)∥_{0,Ω}^{2} + β_{0} ∥u^{(n)}(t) − u(t)∥_{1,Ω}^{2}  (7.2)

+\frac{1}{2}αβ_{0} \frac{d}{dt} ∥u^{(n)}(t) − u(t)∥_{1,Ω}^{2}

≤ \frac{1}{2} ∥\overline{f}^{(n)} (t, τ_{h_{n}} \overline{u}^{(n)}(t), ∇τ_{h_{n}} \overline{u}^{(n)}(t)) − f (t, u (t) , ∇u (t))∥_{0,Ω}^{2}

+\frac{1}{2} ∥u^{(n)}(t) − u(t)∥_{0,Ω}^{2}

+κ_{0} ∥\overline{u}^{(n)}(t) − u^{(n)}(t)∥_{1,Ω} ∥u^{(n)}(t) − u(t)∥_{1,Ω} EJQTDE, 2007 No. 14, p. 24 Therefore, performing similar calculations as in Section 6, it comes

∥\overline{f}^{(n)} (t, τ_{h_{n}} \overline{u}^{(n)}, ∇τ_{h_{n}} \overline{u}^{(n)}) − f (t, u (t) , ∇u (t))∥_{0,Ω}^{2} (7.3)
≤ 3L^{2} ((M′ + 6C_{2}^{2}) h_{n}^{2} + ∥u^{(n)} (t) − u (t)∥_{1,Ω}^{2}) .

Inserting (7.3) into (7.2), applying the Cauchy inequality and using (5.5)-(5.6), we get

\frac{1}{2} \frac{d}{dt} ∥u^{(n)}(t) − u(t)∥_{0,Ω}^{2} + β_{0} ∥u^{(n)}(t) − u(t)∥_{1,Ω}^{2} (7.4)

+\frac{1}{2}αβ_{0} \frac{d}{dt} ∥u^{(n)}(t) − u(t)∥_{1,Ω}^{2}

≤ \frac{1}{2} (3L^{2} (M′ + 6C_{2}^{2}) + κ_{0}^{2}C_{2}^{2}) h_{n}^{2}

+ (\frac{3L^{2}}{2} + 1) ∥u^{(n)} (t) − u (t)∥_{1,Ω}^{2} .

Integrating (7.4) over (0, t) by taking into account the fact that u^{(n)}(0) = u(0) = u_{0}, and neglecting the first term on the left hand-side, we find

∫_{0}^{t} ∥u^{(n)}(s) − u(s)∥_{1,Ω}^{2} ds + ∥u^{(n)}(t) − u(t)∥_{1,Ω}^{2}

≤ \frac{T}{β_{0} min (2, α)} (3L^{2} (M′ + 6C_{2}^{2}) + κ_{0}^{2}C_{2}^{2}) h_{n}^{2}

+\frac{(3L^{2} + 2)}{β_{0} min (2, α)} ∫_{0}^{t} ∥u^{(n)}(s) − u(s)∥_{1,Ω}^{2} ds, ∀t ∈ \overline{I}.

It then follows, by means of Lemma 2.2

∫_{0}^{t} ∥u^{(n)}(s) − u(s)∥_{1,Ω}^{2} ds + ∥u^{(n)}(t) − u(t)∥_{1,Ω}^{2}  (7.5)
≤ \frac{T}{β_{0} min (2, α)} (3L^{2} (M′ + 6C_{2}^{2}) + κ_{0}^{2}C_{2}^{2}) h_{n}^{2} exp (\frac{(3L^{2} + 2)}{β_{0} min (2, α)}T ) ,

for all t ∈ \overline{I}. Hence, in the left hand-side of (7.5), taking the upper bound with respect to t, we obtain

∥u^{(n)} − u∥_{L^{2}(I,V)}^{2} + ∥u^{(n)} − u∥_{C(\overline{I},V)}^{2}
≤ \frac{T}{β_{0} min (2, α)} (3L^{2} (M′ + 6C_{2}^{2}) + κ_{0}^{2}C_{2}^{2}) h_{n}^{2} exp (\frac{(3L^{2} + 2)}{β_{0} min (2, α)}T ) ,

EJQTDE, 2007 No. 14, p. 25 from which we get estimate (7.1), with

C_{12} = \sqrt{\frac{T (3L^{2} (M′ + 6C_{2}^{2}) + κ_{0}^{2}C_{2}^{2})}{β_{0} min (2, α)}h_{n}^{2} exp (\frac{(3L^{2} + 2)}{β_{0} min (2, α)}T )}.

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(Received October 10, 2006)

(A. Bouziani and N. Merazga) Département de Mathématiques, Centre Universitaire Larbi Ben M’hidi, Oum El Bouagui, 04000, Algeria.

E-mail address, A. Bouziani: aefbouziani@yahoo.fr

E-mail address, N. Merazga: nabilmerazga@yahoo.fr