ROTHE TIME-DISCRETIZATION METHOD APPLIED
TO A QUASILINEAR WAVE EQUATION SUBJECT
TO INTEGRAL CONDITIONS
ABDELFATAH BOUZIANI AND NABIL MERAZGA
Received 27 January 2004 and in revised form 12 February 2004
This paper presents a well-posedness result for an initial-boundary value problem with
only integral conditions over the spatial domain for a one-dimensional quasilinear wave
equation. The solution and some of its properties are obtained by means of a suitable
application of the Rothe time-discretization method.
1. Introduction
Recently, the study of initial-boundary value problems for hyperbolic equations with
boundary integral conditions has received considerable attention. This kind of conditions has many important applications. For instance, they appear in the case where a
direct measurement quantity is impossible; however, their mean values are known.
In this paper, we deal with a class of quasilinear hyperbolic equations (T is a positive
constant):
∂v
∂2 v ∂2 v
−
= f x,t,v,
,
∂t 2 ∂x2
∂t
(x,t) ∈ (0,1) × [0,T],
(1.1)
subject to the initial conditions
v(x,0) = v0 (x),
∂v
(x,0) = v1 (x),
∂t
0 x 1,
(1.2)
and the boundary integral conditions
1
0
1
0
v(x,t)dx = E(t),
0 t T,
xv(x,t)dx = G(t),
0 t T,
(1.3)
where f, v0 , v1 , E, and G are sufficiently regular given functions.
Problems of this type were first introduced in [3], in which the first author proved the
well-posedness of certain linear hyperbolic equations with integral condition(s). Later,
Copyright © 2004 Hindawi Publishing Corporation
Advances in Difference Equations 2004:3 (2004) 211–235
2000 Mathematics Subject Classification: 35L05, 35D05, 35B45, 35B30
URL: http://dx.doi.org/10.1155/S1687183904401071
212
On a quasilinear wave equation with integral conditions
similar problems have been studied in [1, 4, 5, 6, 7, 8, 16, 24, 25] by using the energetic
method, the Schauder fixed point theorem, Galerkin method, and the theory of characteristics. We refer the reader to [2, 9, 10, 11, 12, 13, 14, 15, 17, 21, 22, 23, 26] for other
types of equations with integral conditions.
Differently to these works, in the present paper, we employ the Rothe time-discretization method to construct the solution. This method is a convenient tool for both the
theoretical and numerical analyses of the stated 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 solution. We remark that the application of Rothe method to
this nonlocal problem is made possible thanks to the use of the so-called Bouziani space,
first introduced by the first author, see, for instance, [4, 6, 20].
Introducing a new unknown function u(x,t) = v(x,t) − r(x,t), where
r(x,t) = 6 2G(t) − E(t) x − 2 3G(t) − 2E(t) ,
(1.4)
problem (1.1)–(1.3) with inhomogeneous integral conditions (1.3) can be equivalently
reduced to the problem of finding a function u satisfying
∂u
∂2 u ∂2 u
− 2 = f x,t,u,
, (x,t) ∈ (0,1) × I,
2
∂t
∂x
∂t
∂u
u(x,0) = U0 (x),
(x,0) = U1 (x), 0 x 1,
∂t
1
0
1
0
(1.5)
(1.6)
u(x,t)dx = 0,
t ∈ I,
(1.7)
xu(x,t)dx = 0,
t ∈ I,
(1.8)
where
f x,t,u,
I := [0,T],
∂u
∂u ∂r
∂2 r
:= f x,t,u + r,
+
− 2,
∂t
∂t ∂t
∂t
U0 (x) := v0 (x) − r(x,0),
U1 (x) = v1 (x) −
(1.9)
∂r
(x,0).
∂t
Hence, instead of looking for v, we simply look for u. The solution of problem (1.1)–(1.3)
will be directly obtained by the relation v = u + r.
The paper is divided as follows. In Section 2, we present notations, definitions, assumptions, and some auxiliary results. Moreover, the concept of the required solution is
stated, as well as the main result of the paper. Section 3 is devoted to the construction
of approximate solutions of problem (1.5)–(1.8) by solving the corresponding linearized
time-discretized problems, while in Section 4, some a priori estimates for the approximations are derived. We end the paper by Section 5 where we prove the convergence of the
method and the well-posedness of the investigated problem.
A. Bouziani and N. Merazga 213
2. Preliminaries, notation, and main result
Let H 2 (0,1) be the (real) second-order Sobolev space on (0,1) with norm · H 2 (0,1) and
let (·, ·) and · be the usual inner product and the corresponding norm, respectively,
in L2 (0,1). The nature of the boundary conditions (1.7) and (1.8) suggests introducing
the following space:
2
V := φ ∈ L (0,1);
1
0
φ(x)dx =
1
0
xφ(x)dx = 0 ,
(2.1)
which is clearly a Hilbert space for (·, ·).
Our analysis requires the use of the so-called Bouziani space B21 (0,1) (see, e.g., [4, 5])
defined as the completion of the space C0 (0,1) of real continuous functions with compact
support in (0,1), for the inner product
(u,v)B21 =
1
0
x u · x v dx
(2.2)
and the associated norm
v B21 = (v,v)B21 ,
where x v :=
inequality
x
0
(2.3)
v(ξ)dξ for every fixed x ∈ (0,1). We recall that, for every v ∈ L2 (0,1), the
v 2B1 2
1
v 2
2
(2.4)
holds, implying the continuity of the embedding L2 (0,1) B21 (0,1).
Moreover, we will work in the standard functional spaces of the types C(I,X),
C 0,1 (I,X), L2 (I,X), and L∞ (I,X), where X is a Banach space, the main properties of which
can be found in [19].
For a given function w(x,t), the notation w(t) is automatically used for the same function considered as an abstract function of the variable t ∈ I into some functional space
on (0,1). Strong or weak convergence is denoted by → or , respectively.
The Gronwall lemma in the following continuous and discrete forms will be very useful to us thereafter.
Lemma 2.1. (i) Let x(t) 0, and let h(t), y(t) be real integrable functions on the interval
[a,b]. If
y(t) h(t) +
t
a
x(τ)y(τ)dτ,
∀t ∈ [a,b],
(2.5)
then
y(t) h(t) +
t
a
t
h(τ)x(τ)exp
τ
x(s)ds dτ,
∀t ∈ [a,b].
(2.6)
214
On a quasilinear wave equation with integral conditions
In particular, if x(τ) ≡ C is a constant and h(τ) is nondecreasing, then
y(t) h(t)eC(t−a) ,
∀t ∈ [a,b].
(2.7)
(ii) Let {ai } be a sequence of real nonnegative numbers satisfying
i
ai a + bh
ak ,
∀i = 1,...,
(2.8)
k =1
where a, b, and h are positive constants with h < 1/b. Then
a
b(i − 1)h
,
ai exp
1 − bh
1 − bh
∀i = 1,2,....
(2.9)
Proof. The proof is the same as that of [18, Lemma 1.3.19].
Throughout the paper, we will make the following assumptions:
(H1 ) f (t,w, p) ∈ L2 (0,1) for each (t,w, p) ∈ I ×V × V and the following Lipschitz
condition:
f (t,w, p) − f (t ,w , p ) 1 l |t − t | + w − w 1 + p − p 1
B2
B2
B2
(2.10)
is satisfied for all t,t ∈ I and all w,w , p, p ∈ V , for some positive constant l;
(H2 ) U0 ,U1 ∈ H 2 (0,1);
(H3 ) the compatibility condition U0 ,U1 ∈ V , that is, concretely,
1
0
1
0
U0 (x)dx =
U1 (x)dx =
1
0
1
0
xU0 (x)dx = 0,
(2.11)
xU1 (x)dx = 0.
(2.12)
We look for a weak solution in the following sense.
Definition 2.2. A weak solution of problem (1.5)–(1.8) means a function u : I → L2 (0,1)
such that
(i) u ∈ C 0,1 (I,V );
(ii) u has (a.e. in I) strong derivatives du/dt ∈ L∞ (I,V ) ∩ C 0,1 (I,B21 (0,1)) and
d2 u/dt 2 ∈ L∞ (I,B21 (0,1));
(iii) u(0) = U0 in V and (du/dt)(0) = U1 in B21 (0,1);
(iv) the identity
d2 u
(t),φ
dt 2
du
+ u(t),φ = f t,u(t), (t) ,φ
1
dt
B2
holds for all φ ∈ V and a.e. t ∈ I.
B21
(2.13)
A. Bouziani and N. Merazga 215
Note that since u ∈ C 0,1 (I,V ) and du/dt ∈ C 0,1 (I,B21 (0,1)), condition (iii) makes sense,
whereas assumption (H1 ), together with (i) and the fact that du/dt ∈ L∞ (I,V ) and
d2 u/dt 2 ∈ L∞ (I,B21 (0,1)), implies that (2.13) is well defined. On the other hand, the fulfillment of the integral conditions (1.7) and (1.8) is included in the fact that u(t) ∈ V , for
all t ∈ I.
The main result of the present paper reads as follows.
Theorem 2.3. Under assumptions (H1 ), (H2 ), and (H3 ), problem (1.5)–(1.8) admits a
unique weak solution u, in the sense of Definition 2.2, that depends continuously upon the
data f , U0 , and U1 . Moreover, the following convergence statements hold:
1
u −→ u in C(I,V ), with convergence order O 1/2 ,
n
du
δun −→
in C I,B21 (0,1) ,
dt
d2 u
d n
δu 2 in L2 I,B21 (0,1) ,
dt
dt
n
(2.14)
as n→∞, where the sequences {un }n and {δun }n are defined in (3.18) and (3.19), respectively.
3. Construction of an approximate solution
Let n be an arbitrary positive integer, and let {t j }nj=1 be the uniform partition of I, t j = jhn
with hn = T/n. Successively, for j = 1,...,n, we solve the linear stationary boundary value
problem
u j − 2u j −1 + u j −2 d2 u j
−
= fj,
h2n
dx2
1
0
1
0
x ∈ (0,1),
(3.1)
u j (x)dx = 0,
(3.2)
xu j (x)dx = 0,
(3.3)
where
f j := f t j ,u j −1 ,
u j −1 − u j −2
,
hn
(3.4)
starting from
u−1 (x) = U0 (x) − hn U1 (x),
u0 (x) = U0 (x),
x ∈ (0,1).
(3.5)
Lemma 3.1. For each n ∈ N∗ and each j = 1,...,n, problem (3.1) j –(3.3) j admits a unique
solution u j ∈ H 2 (0,1).
Proof. We use induction on j. For this, suppose that u j −1 and u j −2 are already known
and that they belong to H 2 (0,1), then f j ∈ L2 (0,1). From the classical theory of linear
ordinary differential equations with constant coefficients, the general solution of (3.1) j
216
On a quasilinear wave equation with integral conditions
which can be written in the form
−2u j −1 + u j −2
d2 u j
1
− 2 uj =
− fj
2
dx
hn
h2n
(3.6)
is given by
u j (x) = k1 (x)cosh
x
x
+ k2 (x)sinh ,
hn
hn
x ∈ (0,1),
(3.7)
where k1 and k2 are two functions of x satisfying the linear algebraic system
dk1
x dk
x
(x)cosh + 2 (x)sinh = 0,
dx
hn dx
hn
x dk
x
dk1
(x)sinh + 2 (x)cosh = hn F j (x),
dx
hn dx
hn
(3.8)
with
F j :=
−2u j −1 + u j −2
h2n
− fj.
(3.9)
Since the determinant of (3.8) is
∆ = cosh2
x
2 x
− sinh
= 1,
hn
hn
(3.10)
then
0
dk1
(x) = dx
hn F j (x)
x
cosh
dk2
h
n
(x) = dx
sinh x
h
x hn = −hn F j (x)sinh x ,
x
hn
cosh hn sinh
= hn F j (x)cosh x ,
hn
hn F j (x)
(3.11)
0
n
that is,
x
ξ
dξ + λ1 ,
hn
x
ξ
k2 (x) = hn F j (ξ)cosh dξ + λ2 ,
h
0
n
k1 (x) = −hn
0
F j (ξ)sinh
(3.12)
with λ1 and λ2 two arbitrary real constants. Inserting (3.12) into (3.7), we get
u j (x) = hn
x
0
F j (ξ)sinh
x−ξ
x
x
dξ + λ1 cosh + λ2 sinh .
hn
hn
hn
(3.13)
A. Bouziani and N. Merazga 217
Obviously, the function u j will be a solution to problem (3.1) j –(3.3) j if and only if the
pair (λ1 ,λ2 ) is selected in such a manner that conditions (3.2) j and (3.3) j hold, that is,
1
1
1 x
x
x
x−ξ
dx + λ2 sinh dx = −hn
F j (ξ)sinh
dξ dx,
hn
hn
hn
0
0
0 0
1
1
1x
x
x
x−ξ
xF j (ξ)sinh
dξ dx.
λ1 x cosh dx + λ2 x sinh dx = −hn
hn
hn
hn
0
0
0 0
λ1
cosh
(3.14)
An easy computation shows that (λ1 ,λ2 ) is the solution of the linear algebraic system
1 x
1
1
x−ξ
+ λ2 cosh − 1 = −
F j (ξ)sinh
dξ dx,
hn
hn
hn
0 0
1
1
1
1
λ1 sinh − hn cosh + hn + λ2 cosh − hn sinh
hn
hn
hn
hn
1x
x−ξ
xF j (ξ)sinh
dξ dx,
=−
hn
0 0
λ1 sinh
(3.15)
whose determinant is
1
1
+ sinh
hn
hn
1
1
1
= 2sinh
cosh
− 2hn sinh
.
2hn
2hn
2hn
D hn = 2hn − 2hn cosh
(3.16)
Note that D(hn ) does not vanish for any hn > 0, indeed equation D(hn ) = 0 is equivalent
to the equation cosh(1/2hn ) − 2hn sinh(1/2hn ) = 0, that is, to the equation tanh(1/2hn ) =
1/2hn which clearly has no solution. Therefore, for all hn > 0, system (3.15) admits a
unique solution (λ1 ,λ2 ) ∈ R2 , which means that problem (3.1) j –(3.3) j is uniquely solv
able, and it is obvious that u j ∈ H 2 (0,1) since F j ∈ L2 (0,1).
Now, we introduce the notations
u j − u j −1
, j = 0,...,n,
hn
δu j − δu j −1 u j − 2u j −1 + u j −2
=
, j = 1,...,n,
δ 2 u j :=
hn
h2n
δu j :=
(3.17)
and construct the Rothe function un : I → H 2 (0,1) ∩ V by setting
un (t) = u j −1 + δu j t − t j −1 ,
t ∈ t j −1 ,t j , j = 1,...,n,
(3.18)
218
On a quasilinear wave equation with integral conditions
and the following auxiliary functions:
δun (t) = δu j −1 + δ 2 u j t − t j −1 ,

u j
un (t) = 
U0

δu j
n
t ∈ t j −1 ,t j , j = 1,...,n,
for t ∈ t j −1 ,t j , j = 1,...,n,
for t ∈ − hn ,0 ,
(3.20)
for t ∈ t j −1 ,t j , j = 1,...,n,
for t ∈ − hn ,0 .
δu (t) = 
U1
(3.19)
(3.21)
We expect that the limit u := limn→∞ un exists in a suitable sense, and that is the desired
weak solution to our problem (1.5)–(1.8). The demonstration of this fact requires some
a priori estimates whose derivation is the subject of the following section.
4. A priori estimates for the approximations
In what follows, c denote generic positive constants which are not necessarily the same at
any two places.
Lemma 4.1. There exist c > 0 and n0 ∈ N∗ such that
u j c,
δu j c,
2 δ u j 1 c,
(4.1)
(4.2)
(4.3)
B2
for all j = 1,...,n and all n n0 .
Proof. To derive these estimates, we need to write problem (3.1) j –(3.3) j in a weak formulation.
Let φ be an arbitrary function from the space V defined in (2.1). One can easily find
that
x
0
(x − ξ)φ(ξ)dξ = 2x φ,
∀x ∈ (0,1),
(4.4)
where
2x φ := x ξ φ =
x
0
ξ
dξ
0
φ(η)dη.
(4.5)
This implies that
21 φ =
1
0
(1 − ξ)φ(ξ)dξ =
1
0
φ(ξ)dξ −
1
0
ξφ(ξ)dξ = 0.
(4.6)
Next, we multiply, for all j = 1,...,n, (3.1) j by 2x φ and integrate over (0,1) to get
1
0
δ 2 u j (x)2x φ dx −
1
0
d2 u j
(x)2x φ dx =
dx2
1
0
f j (x)2x φ dx.
(4.7)
A. Bouziani and N. Merazga 219
Here, we used the notations (3.17). Performing some standard integrations by parts for
each term in (4.7) and invoking (4.6), we obtain
1
0
δ
2
u j (x)2x φ dx
=
1
0
d 2 2
x δ u j x φ dx
dx
x =1 1
= x δ 2 u j 2x φ
− x δ 2 u j x φ dx
0
x =0
= − δ 2 u j ,φ B21 ,
x =1 1
1 2
d uj
du j
du j
2
2 (x)x φ dx =
(x)x φ −
(x)x φ dx
2
dx
0
dx
=−
1
0
x =0
0
dx
du j
(x)x φ dx
dx
(4.8)
x =1 1
= −u j (x)x φ
+ u j (x)φ(x)dx
0
x =0
= u j ,φ ,
1
1
d
f j (x)2x φ dx =
x f j 2x φ dx
0
0
dx
x =1 1
= x f j 2x φ
− x f j x φ dx
0
x =0
= − f j ,φ B21 ,
so that (4.7) becomes finally
δ 2 u j ,φ
B21
+ u j ,φ = f j ,φ
B21 ,
∀φ ∈ V , ∀ j = 1,...,n.
(4.9)
Now, for i = 2,..., j, we take the difference of the relations (4.9)i − (4.9)i−1 , tested with
φ = δ 2 ui = (δui − δui−1 )/hn which belongs to V in view of (3.2)i − (3.3)i , (3.2)i−1 −
(3.3)i−1 , and (H3 ). We have
δ 2 ui − δ 2 ui−1 ,δ 2 ui
B21
+ δui ,δui − δui−1 = fi − fi−1 ,δ 2 ui
B21 ,
(4.10)
then, using the identity
2 v,v − w = v2 − w2 + v − w2
(4.11)
and its analog for (·, ·)B21 , it follows that
2 2
δ ui 1 − δ 2 ui−1 2 1 + δ 2 ui − δ 2 ui−1 2 1 + δui 2
B2
B2
B2
2 2
− δui−1 + δui − δui−1 = 2 fi − fi−1 ,δ 2 ui B21 ,
(4.12)
hence, omitting the third and last terms in the left-hand side, we get
2 2 2 2
δ ui 1 + δui δ ui−1 2 1 + δui−1 2 + 2 fi − fi−1 1 δ 2 ui 1 .
B2
B2
B2
B2
(4.13)
220
On a quasilinear wave equation with integral conditions
We sum up these inequalities and obtain
j
2 2 δ u j 1 + δu j 2 δ 2 u1 2 1 + δu1 2 + 2 fi − fi−1 1 δ 2 ui 1 ,
B2
B2
B2
B2
(4.14)
i=2
hence, thanks to the Cauchy inequality
1
2ab a2 + εb2 ,
ε
∀a,b ∈ R, ∀ε ∈ R∗
+,
(4.15)
we can write, for ε = hn ,
j
j
2 2 δ u j 1 + δu j 2 δ 2 u1 2 1 + δu1 2 + 1
f i − f i −1 2 1 + h n δ 2 u i 2 1 .
B2
B2
B2
B2
hn i=2
To majorize
j
2
i=2 fi − fi−1 B21 ,
(4.16)
i =2
we remark that
fi − fi−1 2 1 = f ti ,ui−1 ,δui−1 − f ti−1 ,ui−2 ,δui−2 2 1
B2
B2
2
l2 hn + ui−1 − ui−2 1 + δui−1 − δui−2 1
B2
2
= l2 h2n 1 + δui−1 B21 + δ 2 ui−1 B21
2 2 3l2 h2 1 + δui−1 1 + δ 2 ui−1 1 ,
n
B2
B2
B2
(4.17)
i = 2,..., j.
Summing up for i = 2,..., j, we may arrive at
j
j fi − fi−1 2 1 3l2 ( j − 1)h2 + 3l2 h2
δui−1 2 1 + δ 2 ui−1 2 1
n
n
B2
B2
B2
i=2
(4.18)
i=2
or
j
j −1 2 2 2 fi − fi−1 2 1 3l2 ( j − 1)h2 + 3l2 h2
δ ui 1 + δui 1 .
n
n
B2
B2
B2
i=2
(4.19)
i=1
To estimate δ 2 u1 2B21 + δu1 2 , we test the relation (4.9)1 with φ = δ 2 u1 = (δu1 − δu0 )/hn
= (δu1 − U1 )/hn which is an element of V owing to (3.2)1 –(3.3)1 and assumption (H3 ).
We have
2 2
δ u1 1 + u1 ,δu1 − U1 = f1 ,δ 2 u1 1
B2
B2
hn
(4.20)
or
2 2 δ u1 1 + δu1 ,δu1 − U1 = f1 ,δ 2 u1
B2
B21
− U0 ,δ 2 u1 .
(4.21)
A. Bouziani and N. Merazga 221
But
1
d
x δ 2 u1 dx
dx
0
x =1 1
dU0
= U0 (x)x δ 2 u1 −
(x)x δ 2 u1 dx
0 dx
x =0
1
dU0
=−
(x)x δ 2 u1 dx,
0 dx
U0 ,δ 2 u1 =
U0 (x)
(4.22)
and since
x
d2 U0
dU0
dU0
(x) −
(0),
=
2
dx
dx
dx
∀x ∈ (0,1),
(4.23)
we get, due to (4.6),
1
d2 U0
dU0
(0)21 δ 2 u1
x δ 2 u1 dx −
dx2
dx
0
1 2 d U0
= − x
x δ 2 u1 dx
dx2
0
2
d U0 2
=−
,δ
u
,
1
dx2
B21
U0 ,δ 2 u1 = −
x
(4.24)
in light of which (4.21) becomes
2 2 δ u1 1 + δu1 ,δu1 − U1 =
B2
f1 +
d2 U0 2
,δ u1
dx2
B21
.
(4.25)
Therefore,
2
2 2
δ u1 1 + 1 δu1 2 − 1 U1 2 + 1 δu1 − U1 2 f1 + d U0 δ 2 u1 1 ,
B2
B2
2
2
2
dx2 1
(4.26)
B2
hence,
2
2
2
2δ 2 u1 B21 + δu1 U1 + 2
f1 +
d2 U0 δ 2 u 1 1
B2
2
1
dx B2
2
2 2 2
d2 U0 U1 + f1 + dx2 1 + δ u1 B21
B2
(4.27)
2 d 2 U0 2
2
2
U1 + 2 f1 B21 + 2 + δ 2 u1 B21 ,
dx 1
B2
from which it follows that
2 2 2 2 δ u1 1 + δu1 2 U1 2 + 2 c1 + d U0 ,
B2
2 dx
B21
(4.28)
222
On a quasilinear wave equation with integral conditions
where c1 := maxt∈I f (t,U0 ,U1 )2B21 < ∞ in virtue of (H1 ). Substituting (4.19) and (4.28)
in (4.16), this gives
2 2 2 2 δ u j 1 + δu j 2 U1 2 + 2 c1 + d U0 + 3l2 ( j − 1)hn
B2
2 dx
B21
j −1 j
2 2
2 2
+ 3l hn
δ ui B21 + δui B21 + hn δ 2 ui B21
2
i =1
i=2
2 2 2
d U0 2
U1 + 2 c1 + dx2 1 + 3l ( j − 1)hn
B2
j 2 2 2 δ 2 ui 2 1 + δui 2 + hn
δ ui 1 + δui + 3l2 hn
j
B2
i =1
i=1
(4.29)
B2
2 2 d U0 2
2
= U1 + 2 c1 + dx2 1 + 3l ( j − 1)hn
B2
δ 2 ui 2 1 + δui 2 ;
j
+ 3l2 + 1 hn
B2
i =1
consequently,
2 2 2 2 δ u j 1 + δu j 2 U1 2 + 2 c1 + d U0 + 3l2 T
B2
2 dx
B21
j 2 2 2 δ ui 1 + δui ,
+ 3l + 1 hn
B2
2
(4.30)
∀ j = 1,...,n.
i=1
By the discrete Gronwall lemma, we conclude that
2 2 δ u j 1 + δu j 2
B2
2
U1 + 2 c1 + d 2 U0 /dx2 2 1 + 3l2 T
B2
2
2
e(3l +1)( j −1)hn /(1−(3l +1)hn ) ,
1 − 3l2 + 1 hn
(4.31)
for all j = 1,...,n, provided that hn < 1/(3l2 + 1). But, since hn is intended to tend towards
zero, we can, without loss of generality, consider that hn 1/2(3l2 + 1) with hn T of
course. In this case, inequality (4.31) implies, for all j = 1,...,n, that
2 2 2 2 2
2
δ u j 1 + δu j 2 2 U1 2 + 2 c1 + d U0 + 3l T e2(3l +1)T
B2
2 dx
B21
(4.32)
if 1/2(3l2 + 1) T, and
2
2
2 2
2
2 2 2 U1 + 2 c1 + d U0 /dx B21 + 3l T (3l2 +1)T/2(1−(3l2 +1)T)
δ u j 1 + δu j e
(4.33)
B2
1 − 3l2 + 1 T
A. Bouziani and N. Merazga 223
otherwise. Estimates (4.2) and (4.3) then follow with
 2 2 

2
d U0 1

2

2

2 U1 + 2 c1 + 2 + 3l T e(3l +1)T , if T 2 ,


1

dx B2
2 3l +1





c := c2 := 
U1 2 + 2 C1 + d 2 U0 /dx2 2B1 + 3l2 T
2
2
2


e(3l +1)T/2(1−(3l +1)T) ,


2 +1 T

1
−
3l



1



,
if T < 2
2 3l + 1
(4.34)
for all n n0 , where n0 is any positive integer such that T/n0 1/2(3l2 + 1), that is, n0 2T(3l2 + 1).
Finally, from the identity
u j = U0 + hn
j
δui ,
∀ j = 1,...,n,
(4.35)
i=1
we deduce in light of what precedes that
j
u j U0 + hn δui U0 + jhn c2 ,
(4.36)
i=1
hence
u j U0 + Tc2 := c,
∀ j = 1,...,n,
(4.37)
which finishes the proof.
As a consequence of Lemma 4.1, we have the following corollary.
Corollary 4.2. There exist c > 0 such that the estimates
n du
dt (t) c,
n
n
u (t) − un (t) chn ,
u (t) − un t − hn chn ,
d
n n δun (t) c,
δu (t) c,
δu (t) c,
dt
1
B2
n
δu (t) − δun (t) 1 chn ,
δun (t) − δun t − hn 1 chn ,
B2
B2
n dun δu −
chn
n u (t) c,
n u (t) c,
dt
hold for all t ∈ I and n n0 .
L2 (I,B21 )
(4.38)
(4.39)
(4.40)
(4.41)
(4.42)
224
On a quasilinear wave equation with integral conditions
Proof. Obviously, estimates (4.38)1 and (4.38)2 are a direct consequence of (4.1), while
estimates (4.40)1 and (4.40)2 follow immediately from (4.2). On the other hand, since

δu j ,
dun
(t) = 
dt
δu1 ,
∀t ∈ t j −1 ,t j , 1 j n,
t = 0,

δu j t j − t , ∀t ∈ t j −1 ,t j , 1 j n,
un (t) − un (t) = 
0,
t = 0,

δu j t − t j −1 , ∀t ∈ t j −1 ,t j , 1 j n,
n
n
u (t) − u t − hn = 
0,
t = 0,
(4.43)
we derive
n du
δu j ,
dt (t) 1max
j n
n
u (t) − un (t) hn max δu j ,
(4.44)
1 j n
n
u (t) − un t − hn hn max δu j ,
1 j n
from which estimates (4.38)3 and (4.39) follow, thanks to (4.2). Similarly, from the identities

∀t ∈ t j −1 ,t j , 1 j n,
δ 2 u j ,
d n
δu (t) =  2
dt
δ u1 ,
t = 0,

δ 2 u j t j − t , ∀t ∈ t j −1 ,t j , 1 j n,
n
n
δu (t) − δu (t) = 
0,
t = 0,

δ 2 u j t − t j −1 , ∀t ∈ t j −1 ,t j , 1 j n,
n
n
δu (t) − δu t − hn = 
0,
t = 0,
δun (t) −
dun
(t) = δ 2 u j t − t j ,
dt
(4.45)
∀t ∈ t j −1 ,t j , 1 j n,
we deduce the remaining estimates (4.40)3 , (4.41), and (4.42) in view of (4.3).
5. Convergence and existence result
We define, for all n n0 , the abstract function f
n
(n)
: I × V × V → L2 (0,1) by
∀t ∈ t j −1 ,t j , j = 1,...,n.
f (t,w, p) = f t j ,w, p ,
(5.1)
Then the variational equations (4.9) j may be written anew as
d n
δu (t),φ
dt
B21
+ un (t),φ = f
for all φ ∈ V and all t ∈ (0,T].
n
n
t,un t − hn ,δu t − hn ,φ
B21 ,
(5.2)
A. Bouziani and N. Merazga 225
Before we pass to the limit n → ∞ in the approximation scheme (5.2)n , we must establish some convergence assertions.
Theorem 5.1. There exists a function u ∈ C 0,1 (I,V ) with du/dt ∈ L∞ (I,V ) ∩ C 0,1 (I,B21 (0,1))
and d2 u/dt 2 ∈ L∞ (I,B21 (0,1)) such that
(i) un → u in C(I,V );
(ii) un (t) → u(t) in V for all t ∈ I;
(iii) δun → du/dt in C(I,B21 (0,1));
n
(iv) δu du/dt in V for all t ∈ I;
(v) dun /dt du/dt in L2 (I,V );
(vi) (d/dt)δun d2 u/dt 2 in L2 (I,B21 (0,1)).
Moreover, the error estimate
n du C(I,V ) + δu −
dt n
u − u
C(I,B21 )
ch1/2
n
(5.3)
takes place for all n n0 .
Proof. The key point to the proof is to show that {un }n and {δun }n are Cauchy sequences in the Banach spaces C(I,V ) and C(I,B21 (0,1)), respectively. For this, we consider the Rothe functions (3.18) un and um corresponding to the step lengths hn = T/n
n,m
n
m
and hm = T/m, respectively, with m > n n0 . Putting φ = δu (t) := δu (t) − δu (t) in
the difference (5.2)n −(5.2)m , we get, for all t ∈ (0,T],
n,m
d n
δu (t) − δum (t) ,δu (t)
dt
B21
+ un (t) − um (t),δu
n,m
n
m
n,m
(t) = f − f ,δu
(t)
B21 ,
(5.4)
where the abbreviation
n
f := f
n
n
t,un t − hn ,δu t − hn
(5.5)
has been used. Observing that
un − um = un − un + un − um + um − um ,
n
dun
(t) = δu (t),
dt
∀t ∈ (0,T],
(5.6)
we can write
un (t) − um (t),δu
n,m
(t) =
n,m
un (t) − un (t) + um (t) − um (t) ,δu
(t)
d n
+ u (t) − u (t),
u (t) − um (t)
dt
n,m
= un (t) − un (t) + um (t) − um (t) ,δu (t)
n
+
m
1 d
un (t) − um (t)2 ,
2 dt
for a.e. t ∈ I.
(5.7)
226
On a quasilinear wave equation with integral conditions
Analogously, we have
n,m
d n
δu (t) − δum (t) ,δu (t)
dt
B21
n
m
m
d n
m
n
=
δu (t) − δu (t) , δu (t) − δu (t) + δu (t) − δu (t)
dt
B21
d n
+
δu (t) − δum (t) ,δun (t) − δum (t)
dt
B21
n
m
d
=
δun (t) − δum (t) , δu (t) − δun (t) + δum (t) − δu (t)
dt
B21
1 d n
2
+
δu (t) − δum (t)B21 , for a.e. t ∈ I.
2 dt
(5.8)
Substituting (5.7) and (5.8) in (5.4) and rearranging, we obtain
1 d
un (t) − um (t)2 + 1 d δun (t) − δum (t)2 1
B2
2 dt
2 dt
m
n
d n
=
δu (t) − δum (t) , δun (t) − δu (t) + δu (t) − δum (t)
dt
B21
+
un (t) − un (t) + um (t) − um (t) ,δu
n,m
n
m
(t) + f − f ,δu
n,m
(t)
(5.9)
B21 .
Estimating the first two terms in the right-hand side, we write
m
n
d n
δu (t) − δum (t) , δun (t) − δu (t) + δu (t) − δum (t)
dt
d
n
+ d δum (t)
(t)
δu
1 dt
1
dt
B2
B2
n
m
× δun (t) − δu (t)B21 + δu (t) − δum (t)B21
B21
(5.10)
c hn + hm
in view of (4.40)3 and (4.41)1 . Similarly,
un (t) − un (t) + um (t) − um (t) ,δu
n,m
(t)
n
m
un (t) − un (t) + um (t) − um (t) δu (t) + δu (t)
c hn + hm
(5.11)
in view of (4.39)1 and (4.40)2 . It remains to dominate the last term in the right-hand side
in (5.9). For any t fixed in (0,T], there exist two integers k and i corresponding to the
subdivision of I into n and m subintervals, respectively, such that t ∈ (tk−1 ,tk ] ∩ (ti−1 ,ti ].
A. Bouziani and N. Merazga 227
Consequently, owing to assumption (H1 ), it follows that
n
f − f m 1
B2
n
n
m
m
= f tk ,un t − hn ,δu t − hn − f ti ,um t − hm ,δu t − hm B21
n
m
l tk − ti + un t − hn − um t − hm B21 + δu t − hn − δu t − hm B21
l hn + hm + un t − hn − un (t)B21 + un (t) − um (t)B21 + um (t) − um t − hm B21
n
m
+ δu t − hn − δun (t)B21 + δun (t) − δum (t)B21 + δum (t) − δu t − hm B21 ,
(5.12)
then, according to (4.39)2 and (4.41)2 , we have
n
f − f m 1 c hn + hm + l un (t) − um (t) 1 + δun (t) − δum (t) 1 .
B2
B2
B2
(5.13)
On the other hand, due to (4.41)1 , we estimate
n,m δu (t) 1
B2
n
m
δu (t) − δun (t)B21 + δun (t) − δum (t)B21 + δum (t) − δu (t)B21
c hn + hm + δun (t) − δum (t) 1 .
(5.14)
B2
From (5.13) and (5.14), we conclude, thanks to (4.40)1 and (4.38)1 , that
n
m
n,m
f − f ,δu
(t)
B21
n
m n,m
f − f B21 δu (t)B21
2
c hn + hm + c hn + hm δun (t) − δum (t)B21
+ c hn + hm un (t) − um (t)B21 + δun (t) − δum (t)B21
2
+ lun (t) − um (t)B21 δun (t) − δum (t)B21 + lδun (t) − δum (t)B21
2
c hn + hm + c hn + hm δun (t)B21 + δum (t)B21
+ c hn + hm un (t) 1 + um (t) 1 + δun (t) 1 + δum (t) 1
B2
B2
B2
(5.15)
B2
2 2 2
l + un (t) − um (t)B21 + δun (t) − δum (t)B21 + lδun (t) − δum (t)B21 ;
2
here, the elementary inequality ab (1/2)(a2 + b2 ) has been used. Hence,
n
m
f − f ,δu
n,m
(t)
c hn + hm
2
B21
2
2 (5.16)
l
3l + c hn + hm + un (t) − um (t)B21 + δun (t) − δum (t)B21 .
2
2
228
On a quasilinear wave equation with integral conditions
Combining (5.9), (5.10), (5.11), and (5.16), we obtain for a.e. t ∈ I,
d
un (t) − um (t)2 + d δun (t) − δum (t)2 1
B2
dt
dt
2
2
2
n
c hn + hm + c hn + hm + lu (t) − um (t) + 3lδun (t) − δum (t)B21 .
(5.17)
Integrating over (0,t) with consideration to the fact that
un (0) = um (0) = U0 ,
(5.18)
δun (0) = δum (0) = U1 ,
we have
n
u (t) − um (t)2 + δun (t) − δum (t)2 1
B2
t
2
2
c hn + hm + c hn + hm + l un (τ) − um (τ) dτ
t
+ 3l
0
0
n
δu (τ) − δum (τ)2 1 dτ,
B2
(5.19)
∀t ∈ I,
or, by Gronwall’s lemma,
n
u (t) − um (t)2 + δun (t) − δum (t)2 1 c hn + hm 2 + c hn + hm e3lT ,
B2
∀t ∈ I.
(5.20)
Hence, taking the upper bound with respect to t ∈ I in the left-hand side of this last
inequality, we obtain
n
u − u m 2
C(I,V ) +
n
δu − δum 2
C(I,B21 )
c hn + hm
2
+ c hn + hm e3lT ,
(5.21)
from which we deduce that both {un }n and {δun }n are Cauchy sequences in the Banach
spaces C(I,V ) and C(I,B21 (0,1)), respectively. Accordingly, there exist two functions u ∈
C(I,V ) and w ∈ C(I,B21 (0,1)) such that
un −→ u in C(I,V ),
δun −→ w
in C I,B21 (0,1) .
(5.22)
(5.23)
Now, on the basis of estimations (4.38)2 , (4.38)3 , (4.39)1 and the convergence result
(5.22), [18, Lemma 1.3.15] enables us to state the following assertions:
(i) u ∈ C 0,1 (I,V );
(ii) u is strongly differentiable a.e. in I and du/dt ∈ L∞ (I,V );
(iii) un (t) → u(t) in V for all t ∈ I;
(iv) dun /dt du/dt in L2 (I,V ).
On the other hand, in light of estimations (4.40)2 , (4.40)3 , the convergence statement
(5.23), and the continuous embedding V B21 (0,1), [18, Lemma 1.3.15] is also valid
A. Bouziani and N. Merazga 229
n
for the functions δun and the corresponding step functions δu , yielding the following
statements:
(v) w ∈ C 0,1 (I,B21 (0,1));
(vi) w is strongly differentiable a.e. in I and dw/dt ∈ L∞ (I,B21 (0,1));
n
(vii) δu (t) w(t) in V for all t ∈ I;
(viii) (d/dt)δun dw/dt in L2 (I,B21 (0,1)).
We show that w coincides with du/dt. For all v ∈ L2 (I,B21 (0,1)), we have
δun −
du
,v
dt
dun
= δun −
,v
dun du
−
,v
dt
dt
dt
L2 (I,B21 )
L2 (I,B21 )
L2 (I,B21 )
n
n dun du
du
1 +
δu
−
v
−
,v
2
L (I,B2 )
dt L2 (I,B21 )
dt
dt
L2 (I,B21 )
+
(5.24)
or, due to (4.42) and the convergence property (iv) stated above,
δun −
du
,v
dt
L2 (I,B21 )
chn vL2 (I,B21 ) +
dun du
−
,v
dt
dt
L2 (I,B21 )
−→ 0
(5.25)
as n → ∞; hence
δun du
dt
in L2 I,B21 (0,1) ,
(5.26)
which, together with (5.23), yields w = du/dt and consequently dw/dt = d2 u/dt 2 . Finally,
letting m → ∞ in (5.21), taking into account that hn hn0 1/2, we obtain the desired
error estimate. So, the proof is complete.
Now, we are ready to state an existence result.
Theorem 5.2. The limit function u from Theorem 5.1 is the unique weak solution to problem (1.5)–(1.8) in the sense of Definition 2.2.
Proof
Existence. We have to show that the limit function u satisfies all the conditions (i), (ii),
(iii), (iv) of Definition 2.2. Obviously, in light of the properties of the function u listed
in Theorem 5.1, the first two conditions of Definition 2.2 are already seen. On the other
hand, since un → u in C(I,V ) and δun → du/dt in C(I,B21 (0,1)) as n → ∞ and, by construction, un (0) = U0 and δun (0) = U1 , it follows that u(0) = U0 and (du/dt)(0) = U1
hold in V and B21 (0,1), respectively, so the initial conditions (1.6) are also fulfilled, that is,
Definition 2.2(iii) takes place. It remains to see that the integral identity (2.13) is obeyed
by u. For this, we consider the following relation:
n
δu (t) − U1 ,φ
=
t
0
f
t
B21
n
+
0
n
un (τ),φ dτ
n
τ,u τ − hn ,δu τ − hn ,φ
(5.27)
B21
dτ,
∀φ ∈ V , ∀t ∈ I,
230
On a quasilinear wave equation with integral conditions
which results from (5.2)n by integration between 0 and t ∈ I, noting that δun (0) = U1 .
First, by virtue of Theorem 5.1(iii), we have
δun (t) − U1 ,φ
B21
−→
n→∞
du
(t) − U1 ,φ
dt
B21
,
∀φ ∈ V , ∀t ∈ I.
(5.28)
Next, according to estimate (4.38)2 , the expression |(un (τ),φ)| is uniformly bounded with
respect to both n and τ, so the Lebesgue theorem of dominated convergence may be
applied to the convergence statement (ii) from Theorem 5.1, yielding
t
0
un (τ),φ dτ −→
t
n→∞
0
∀φ ∈ V , ∀t ∈ I.
u(τ),φ dτ,
(5.29)
To investigate the behavior of the right-hand side of (5.27) as n → ∞, we first observe that
for all τ ∈ (t j −1 ,t j ], 1 j n, we have
n
f τ,un τ − hn ,δun τ − hn − f τ,u(τ), du (τ) 1
dt
B2
n
du
n
=
(τ) f t j ,u τ − hn ,δu τ − hn − f τ,u(τ),
dt
l t j − τ + un τ − hn − u(τ)
(5.30)
B21
n
du
,
(τ)
B21 + δu τ − hn −
dt 1
B2
owing to assumption (H1 ). However, from estimates (4.39)2 and (5.3), we derive
n
u τ − hn − u(τ)
B21
un τ − hn − un (τ)B21 + un (τ) − u(τ)B21
c hn + h1/2
n ,
(5.31)
∀τ ∈ I;
similarly, from estimates (4.41)2 and (5.3), we get
n
δu τ − hn − du (τ)
dt B21
n
n
δu τ − hn − δun (τ)B21 + δu (τ) −
c
hn + h1/2
n
,
du (τ)
dt B21
(5.32)
∀τ ∈ I.
Therefore, for all τ ∈ (0,T], it holds that
n
f τ,un τ − hn ,δun τ − hn − f τ,u(τ), du (τ) dt
B21
c hn + h1/2
n ,
(5.33)
A. Bouziani and N. Merazga 231
from which we deduce that
f
n
n
τ,un τ − hn ,δu τ − hn
du
(τ)
−→ f τ,u(τ),
in B21 (0,1), ∀τ ∈ (0,T].
dt
n→∞
(5.34)
But, inasmuch as
f
n
n
τ,un τ − hn ,δu τ − hn
∀τ ∈ t j −1 ,t j , 1 j n,
= f t j ,u j −1 ,δu j −1 ,
(5.35)
it follows that
n
f τ,un τ − hn ,δun τ − hn 1
B2
max f t j ,u j −1 ,δu j −1 1
B2
1 j n
max f t j ,u j −1 ,δu j −1 − f t j ,0,0 B21 + max f t j ,0,0 B21
1 j n
(5.36)
1 j n
l max u j −1 B21 + δu j −1 B21 + c3 ,
∀τ ∈ (0,T],
1 j n
where c3 := max1 j n f (t j ,0,0)B21 < ∞. Accordingly,
n
f τ,un τ − hn ,δun τ − hn B21
c,
∀τ ∈ (0,T],
(5.37)
in view of estimates (4.1) and (4.2). This shows that
n n
f τ,un τ − hn ,δu τ − hn ,φ B1 (5.38)
2
is uniformly bounded with respect to both n and τ; hence, applying the Lebesgue theorem
to the convergence statement (5.34), we get
t
0
f
n
n
τ,un τ − hn ,δu τ − hn ,φ
B21
dτ −→
n→∞
t 0
f τ,u(τ),
du
(τ) ,φ
dt
B21
dτ,
(5.39)
for all φ ∈ V and all t ∈ I. Finally, performing a limit process n → ∞ in (5.27), taking into
account (5.28), (5.29), and (5.39), we find out that
du
(t) − U1 ,φ
dt
=
t 0
t
B21
+
0
f τ,u(τ),
u(τ),φ dτ
du
(τ) ,φ
dt
(5.40)
B21
dτ,
∀φ ∈ V , ∀t ∈ I.
But the function du/dt : I → B21 (0,1) is strongly differentiable for a.e. t ∈ I, hence, differentiating the just obtained equality with respect to t, we get the desired identity (2.13).
Thus, u weakly solves problem (1.5)–(1.8).
232
On a quasilinear wave equation with integral conditions
Uniqueness. Let u! and u" be two weak solutions of (1.5)–(1.8). From (2.13), for u = u! − u"
and φ = (du/dt)(t), we obtain
du
d2 u
(t), (t)
dt 2
dt
B21
+ u(t),
du
(t)
dt
du!
du"
du
!(t),
(t) − f t, u"(t), (t) , (t)
= f t, u
dt
dt
dt
B21
(5.41)
,
a.e. t ∈ I.
So, with consideration to the fact that (du/dt)(0) = u(0) = 0, integration from 0 to t
yields, in a standard way,
2
1
du (t) + 1 u(t)2
2 dt
2
B21
t du
du!
du"
f τ, u
(τ) dτ
!
(τ),
f
τ,
u
"
(τ),
(τ)
(τ)
−
1
dt
dt
0
B21 dt
B2
t
du
u(τ) 1 + du (τ)
(τ) dτ,
l
B2 1
dt
dt 1
0
t
u(τ)
l
0
B2
2
du
(τ)
dτ,
B21 + dt 1
B2
(5.42)
B2
∀t ∈ I,
due to assumption (H1 ), whence
2
2 t
u(τ)2 + du (τ)
u(t)2 + du (t) 4l
dt 1
dt
1 dτ,
B2
0
B2
∀t ∈ I,
(5.43)
implying, by Gronwall’s lemma, that u(t) = 0, ∀t ∈ I, that is, u! = u", which achieves the
proof.
We terminate the paper by a result of continuous dependence of the solution u upon
data. Concretely, we have the following theorem.
Theorem 5.3. Let u and u∗ be the weak solutions of problem (1.5)–(1.8), corresponding
to (U0 ,U1 , f ) and (U0∗ ,U1∗ , f ∗ ), respectively. Assume that (U0 ,U1 , f ) and (U0∗ ,U1∗ , f ∗ )
satisfy assumptions (H1 ), (H2 ), and (H3 ), then the inequality
2
∗
u(t) − u∗ (t)2 + du (t) − du (t)
dt
1
dt
B2
2
2
U0 − U ∗ + U1 − U ∗ 1
0
+
1
(5.44)
B2
t 2
∗
f τ,u(τ), du (τ) − f ∗ τ,u∗ (τ), du (τ) dτ et
1
dt
dt
0
takes place for all t ∈ I.
B2
A. Bouziani and N. Merazga 233
Proof. Subtracting (2.13) for u and u∗ and putting φ = (du/dt)(t) − (du∗ /dt)(t) in the
resulting relation, we get
d
d
d2 u(t) − u∗ (t) ,
u(t) − u∗ (t)
+ u(t) − u∗ (t),
u(t) − u∗ (t)
2
1
dt
dt
dt
B2
∗
du
du
d
u(t) − u∗ (t)
, a.e. t ∈ I,
(t) − f ∗ t,u∗ (t),
(t) ,
= f t,u(t),
dt
dt
dt
B21
(5.45)
whence
2
1 d
d u(t) − u∗ (t) + 1 d u(t) − u∗ (t)2
2 dt dt
2 dt
B21
d
du
du∗
∗
∗
∗
,
f
t,u(t),
f
t,u
(t),
u(t)
−
u
(t)
(t)
(t)
−
1
1 dt
dt
dt
B2
B2
a.e. t ∈ I.
(5.46)
Then, integrating over (0,t), we have
2
d
2 ∗
∗
dt u(t) − u (t) 1 + u(t) − u (t)
B2
2 2
U0 − U0∗ + U1 − U1∗ B21
t du
du∗
∗
∗
+2 f
τ,u(τ),
f
τ,u
(τ),
−
(τ)
(τ)
dt
dt
0
2 2
U0 − U ∗ + U1 − U ∗ 1
0
1
d
∗
dτ
u(τ)
−
u
(τ)
1
1 dt
B2
B2
B2
t 2
du
du∗
∗
∗
dτ
(τ)
(τ)
+ f
τ,u(τ),
f
τ,u
(τ),
−
1
dt
dt
0
t
d
2
∗
+ u(τ) − u (τ) 1 dτ,
dt
0
B2
B2
∀t ∈ I,
(5.47)
consequently,
2
u(t) − u∗ (t)2 + d u(t) − u∗ (t) dt
1
2 2
U0 − U ∗ + U1 − U ∗ 1
0
1
B2
B2
t 2
du
du∗
∗
∗
+ f τ,u(τ), dt (τ) − f τ,u (τ), dt (τ) 1 dτ
0
B2
2
t d
∗
∗
2 dτ,
+
dt u(τ) − u (τ) 1 + u(τ) − u (τ)
0
(5.48)
B2
for all t ∈ I. Finally, applying the Gronwall lemma, we get inequality (5.44) which repre
sents the continuous dependence of the solution on data.
234
On a quasilinear wave equation with integral conditions
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
S. A. Beilin, Existence of solutions for one-dimensional wave equations with nonlocal conditions,
Electron. J. Differential Equations 2001 (2001), no. 76, 1–8.
N.-E. Benouar and N. I. Yurchuk, A mixed problem with an integral condition for parabolic
equations with a Bessel operator, Differ. Uravn. 27 (1991), no. 12, 2094–2098 (Russian).
A. Bouziani, Mixed problem with integral condition for certain partial differential equations,
Ph.D. thesis, Constantine University, Constantine, Algeria, 1996.
, Solution forte d’un problème mixte avec une condition non locale pour une classe
d’équations hyperboliques [Strong solution of a mixed problem with a nonlocal condition for
a class of hyperbolic equations], Acad. Roy. Belg. Bull. Cl. Sci. (6) 8 (1997), no. 1–6, 53–70
(French).
, Strong solution to a hyperbolic evolution problem with nonlocal boundary conditions,
Maghreb Math. Rev. 9 (2000), no. 1-2, 71–84.
, On the solvability of parabolic and hyperbolic problems with a boundary integral condition, Int. J. Math. Math. Sci. 31 (2002), no. 4, 201–213.
, On initial boundary value problem with Dirichlet integral conditions for a hyperbolic
equation with the Bessel operator, J. Appl. Math. 2003 (2003), no. 10, 487–502.
, A mixed problem with only integral boundary conditions for a hyperbolic equation, Int.
J. Math. Math. Sci. 2004 (2004), no. 24, 1279–1291.
A. Bouziani and N.-E. Benouar, Mixed problem with integral conditions for a third order parabolic equation, Kobe J. Math. 15 (1998), no. 1, 47–58.
J. R. Cannon, The solution of the heat equation subject to the specification of energy, Quart. Appl.
Math. 21 (1963), 155–160.
J. R. Cannon and J. van der Hoek, The existence of and a continuous dependence result for the
solution of the heat equation subject to the specification of energy, Boll. Un. Mat. Ital. Suppl.
1981 (1981), no. 1, 253–282.
, Diffusion subject to the specification of mass, J. Math. Anal. Appl. 115 (1986), no. 2,
517–529.
M. Dehghan, Implicit locally one-dimensional methods for two-dimensional diffusion with a nonlocal boundary condition, Math. Comput. Simulation 49 (1999), no. 4-5, 331–349.
, A finite difference method for a non-local boundary value problem for two-dimensional
heat equation, Appl. Math. Comput. 112 (2000), no. 1, 133–142.
, Fully explicit finite-difference methods for two-dimensional diffusion with an integral
condition, Nonlinear Anal. Ser. A: Theory Methods 48 (2002), no. 5, 637–650.
D. G. Gordeziani and G. A. Avalishvili, Solution of nonlocal problems for one-dimensional oscillations of a medium, Mat. Model. 12 (2000), no. 1, 94–103 (Russian).
N. I. Ionkin, The solution of a certain boundary value problem of the theory of heat conduction
with a nonclassical boundary condition, Differ. Uravn. 13 (1977), no. 2, 294–304 (Russian).
J. Kačur, Method of Rothe in Evolution Equations, Teubner-Texte zur Mathematik, vol. 80, BSB
B. G. Teubner Verlagsgesellschaft, Leipzig, 1985.
A. Kufner, O. John, and S. Fučı́k, Function Spaces, Noordhoff International Publishing, Leyden,
1977.
N. Merazga and A. Bouziani, Rothe method for a mixed problem with an integral condition for
the two-dimensional diffusion equation, Abstr. Appl. Anal. 2003 (2003), no. 16, 899–922.
C. V. Pao, Dynamics of reaction-diffusion equations with nonlocal boundary conditions, Quart.
Appl. Math. 53 (1995), no. 1, 173–186.
, Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary
conditions, J. Comput. Appl. Math. 88 (1998), no. 1, 225–238.
, Numerical solutions of reaction-diffusion equations with nonlocal boundary conditions,
J. Comput. Appl. Math. 136 (2001), no. 1-2, 227–243.
A. Bouziani and N. Merazga 235
[24]
[25]
[26]
L. S. Pulkina, A non-local problem with integral conditions for hyperbolic equations, Electron. J.
Differential Equations 1999 (1999), no. 45, 1–6.
, On the solvability in L2 of a nonlocal problem with integral conditions for a hyperbolic
equation, Differ. Equ. 36 (2000), no. 2, 316–318.
N. I. Yurchuk, A mixed problem with an integral condition for some parabolic equations, Differ.
Uravn. 22 (1986), no. 12, 2117–2126 (Russian).
Abdelfatah Bouziani: Département de Mathématiques, Centre Universitaire Larbi Ben M’hidi,
Oum El Bouaghi 04000, Algeria
E-mail address: af bouziani@hotmail.com
Nabil Merazga: Département de Mathématiques, Centre Universitaire Larbi Ben M’hidi, Oum El
Bouaghi 04000, Algeria
E-mail address: nabilmerazga@yahoo.com