Electronic Journal of Differential Equations, Vol. 2001(2001), No. 21, pp. 1–16.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu ftp ejde.math.unt.edu (login: ftp)
On the solvability of nonlocal pluriparabolic
problems ∗
Abdelfatah Bouziani
Abstract
The aim of this paper is to prove existence, uniqueness, and continuous dependence upon the data of solutions to mixed problems for pluriparabolic equations with nonlocal boundary conditions. The proofs are
based on a priori estimates established in non-classical function spaces
and on the density of the range of the operator generated by the studied
problems.
1
Introduction
In this paper, we study a class of second order pluriparabolic equations with
nonlocal conditions. The aim is to proof existence, uniqueness, and continuous
dependence of generalized solutions.
Evolution problems with nonlocal boundary conditions have received attention in several papers. Most of the papers were directed to second order
parabolic equations, particularly to heat conduction equations; see, for instance,
Cannon et al. [16]-[19], Kamynin [21], Ionkin [20], Yurchuk [27], BenouarYurchuk [1], Bouziani-Benouar [9], Bouziani [2]-[4] and Mesloub-Bouziani [22][23]. For similar problems, related to other equations, we refer the reader to
Bouziani [5]-[8], Bouziani-Benouar [10] and Pulkina [24]-[25]. Mixed problems
with nonlocal boundary conditions or with nonlocal initial conditions were studied in Yurchuk [28], Byszewski et al. [11]-[ 16], and Bouziani [7]-[8].
The presence of integral terms in the boundary conditions can greatly complicate the application of standard functional and numerical techniques. The
main tool used in this paper is the introduction of a new function space in which
we can establish an a priori estimate.
This paper is outlined as follows: In Section 2, we give notation, the statement of two problems, and the basic assumptions. Section 3 is devoted to the
introduction of the function spaces to be used in the rest of the paper. In Section 3, we present abstract formulations of the posed problems and make precise
∗ Mathematics Subject Classifications: 35K70, 35B45, 46E30, 35D05, 35B30.
Key words: Pluriparabolic equation, nonlocal boundary conditions, a priori estimate,
nonclassical function space, generalized solution.
c
2001
Southwest Texas State University.
Submitted December 4, 2000. Published March 26, 2001.
1
2
Nonlocal pluriparabolic problems
EJDE–2001/21
the concept of solution of the problems. In Section 4, we establish a priori estimates which are derived to show the uniqueness and continuous dependence of
the solutions upon the data. Finally, we prove the existence of the solutions in
Section 5.
2
Notation and statement of problems
Throughout this paper we use the following notation:
t = (t1 , . . . , tn ), τ = (τ1 , . . . , τn ), ti,0 = (t1 , . . . , ti−1 , 0, ti+1 , . . . , tn ),
ti,τ = (t1 , . . . , ti−1 , τi , ti+1 , . . . , tn ), ti,T = (t1 , . . . , ti−1 , Ti , ti+1 , . . . , tn ),
dti = dt1 . . . dti−1 dti+1 . . . dtn (i = 1, . . . , n),
Ω =Q(a, b) ⊂ R, with a, b < ∞, Ii = (0, Ti ), where Ti < ∞ (i = 1, . . . , n),
n
I = i=1 Ii ,
∆i = I1 × . . . × Ii−1 × Ii+1 × . . . × In ,
Qi = Ω × I1 × . . . × Ii−1 × Ii+1 × . . . × In ,
Qi,0 = I1 × . . . × Ii−1 × {0} × Ii+1 × . . . × In
We consider the following problem: Given the data σ, p, Φi (i = 1, . . . , n), E, G
and H, find a function v(x, t) satisfying the pluriparabolic equation
(Lv)(x, t) :=
n
X
∂v
1
−
P (t)v = g(x, t),
∂ti
σ(x)
i=1
for (x, t) ∈ Ω × I,
(1)
where
∂
∂v
(p(Jx σ, t) )
∂x
∂x
with Jx σ the primitive of σ, satisfying the conditions
P (t)v =
(`i v)(x, t) := v(x, ti,0 ) = Φi (x, ti ), for (x, t) ∈ Qi,0 (i = 1, . . . , n),
Z
σ(x)v(x, t)dx = E(t) for t ∈ I,
(2)
(3)
Ω
and satisfying one of the conditions
Z
xσ(x)v(x, t)dx = G(t)
for t ∈ I,
(4)
Ω
∂v(a, t)
= H(t) f or t ∈ I.
∂x
(40 )
If σ(x0 ) = 0 for all x0 in Ω, then (1) is called a singular pluriparabolic
equation.
Next we formulate the main assumptions:
EJDE–2001/21
Abdelfatah Bouziani
3
A1: There are positive constants ci , (i = 0, 1, 2, 3) such that
∂p(Jx σ, t) ≤ c2 (i = 1, . . . , n),
c0 ≤ p(Jx σ, t) ≤ c1 , ∂ti
∂p(Jx σ, t) ∂Jx σ ≤ c3 for (x, t) ∈ Ω × I.
A2: x → σ(x) is a positive continuous function on Ω, such that σ(x) ≤ c4 ,
where c4 is a positive constant.
A3: g ∈ C(Ω × I, R), Φi ∈ C 1 (Qi , R) (i = 1, . . . , n), E, G, H ∈ C 1 (I, R),
Z
σ(x)Φi (x, ti )dx = E(ti,0 ), for ti,0 ∈ ∆i (i = 1, . . . , n),
Ω
Z
xσ(x)Φi (x, ti )dx = G(ti,0 ),
for ti,0 ∈ ∆i (i = 1, . . . , n),
Ω
∂Φi (a, ti )
= H(ti,0 ), respectively
∂x
for ti,0 ∈ ∆i (i = 1, . . . , n)
and Φi = Φj (i 6= j; i, j = 1, . . . , n).
Problem (1)-(3) and (4’) can be viewed as a generalization of that in Bouziani
[8], where the author studied a similar problem for the case of a second order
pluriparabolic equation with n = 2, σ(x) = 1, Ω = (0, b) and with the function
p satisfying, in addition of Assumption A1, other supplementary assumptions.
The proof in [8] is based on an a priori estimate, which is established by taking the inner product in L2 -space of the considered equation by the integrodifferential operator
(M u)(x, t1 , t2 ) := 2(b − x)(=x
∂u
∂u
∂u
+ =x
) − p(x, t1 , t2 ) ,
∂t1
∂t2
∂x
where the definition of =x is similar to the definition of =x from Section 3.1.
The results of this paper are also continuations of those obtained by the author
et al. in [2]-[7], [9]-[10] and [22]-[23].
Let us, now, reformulate problem (1)-(4) [respectively, (1)-(3) and (4’)] with
non-homogeneous boundary conditions (3), (4) [respectively, (3), (4’)] as a problem with homogeneous boundary conditions, by introducing a new unknown
function u(x, t) defined as follows:
u(x, t) = v(x, t) − U (x, t),
for (x, t) ∈ Ω × I,
where
U (x, t)
=
E(t)
6(2G(t) − (b + a)E(t))
+
(b − a)σ(x)
(b − a)4 σ(x)
×(3(x − a)2 − 2(b − a)(x − a))
for (x, t) ∈ Ω × I ;
4
Nonlocal pluriparabolic problems
EJDE–2001/21
respectively
U (x, t)
=
(x − a)2
(x − a)H(t) + R
(x − a)2 σ(x)dx
ZΩ
×(E(t) − H(t) (x − a)σ(x)dx) for (x, t) ∈ Ω × I .
Ω
Then, we have to find a function u(x, t), such that
(Lu)(x, t) = g(x, t) − (LU )(x, t) =: f (x, t) for (x, t) ∈ Ω × I,
= Φi (x, ti ) − (`i U )(x, t)
= : ϕi (x, ti )(i = 1, . . . , n)for (x, t) ∈ Qi,0 ,
(`i u)(x, t)
Z
σ(x)u(x, t)dx = 0
for t ∈ I,
(5)
(6)
(7)
Ω
Z
for t ∈ I,
∂u(a, t)
= 0, respectively
for t ∈ I.
∂x
xσ(x)u(x, t)dx = 0
(8)
Ω
3
(80 )
Preliminaries
Function spaces
In this subsection, we introduce and study certain fundamental function spaces.
For this purpose, let us denote by C0 (Ω) the space of the continuous functions
with compact support in Ω. We define on C0 (Ω) the bilinear form ((., .))σ , given
by
Z
((u, θ))σ :=
=x (σu) · =x (σθ)dx,
(9)
Ω
where
=x η :=
Z
x
η(ξ, .)dξ.
a
The bilinear form, defined by (9), is a scalar product in C0 (Ω) for which C0 (Ω)
is not complete. Thus we are led to introduce its completion.
1
Definition 3.1 We denote by B2,σ
(Ω) a completion of C0 (Ω) for the scalar
product defined by (9), called the weighted Bouziani space.
Thus, we have the following result:
1
Proposition 3.1 The space B2,σ
(Ω) is a Hilbert space for the scalar product
1 (Ω) = ((u, θ))σ ,
(u, θ)B2,σ
with the associated norm
kwkB 1
2,σ (Ω)
= k=x (σw)kL2 (Ω) .
EJDE–2001/21
Abdelfatah Bouziani
5
1
Remark 3.1 If σ(x) = 1, then the space B2,1
(Ω) coincides with B21 (Ω), which
was firstly introduced by the author in [4] and [5].
Definition 3.2 Let m be a nonnegative integer and let 1 ≤ p < ∞. We define
m
the space Bp,σ
(Ω) to be a completion of C0 (Ω) in the norm
m
kwkBp,σ
m (Ω) := k=x (σw)kLp (Ω) ,
and for p = 2, we define an inner product by
m
m
m (Ω) := (=
(u, θ)B2,σ
x (σu), =x (σθ))L2 (Ω) .
Definition 3.3 We denote by L2σ (Ω), the Hilbert space of weighted square
integrable functions with the inner product
(u, θ)L2σ (Ω) := (σu, θ)L2 (Ω) ,
and with the associated norm
√ kwkL2σ (Ω) := σwL2 (Ω) .
Let X be a Hilbert space with a norm k.kX .
Definition 3.4 i) We denote by L2 (I, X) the set of all measurable abstract
functions
I 3 t → u(., t) ∈ X
such that
2
kukL2 (I,X) :=
Z
I
2
ku(., t)kX dt < ∞.
ii) Let C(I i , X)(i = 1, . . . , n) be the set of all continuous functions
I i 3 ti → u(. . . , ti , . . .) ∈ X (i = 1, . . . , n),
with
kukC(I i ,X) := max ku(. . . , ti , . . .)kX < ∞ (i = 1, . . . , n).
ti ∈I i
Proposition 3.2 i) L2 (I, X) is a Hilbert space.
ii) C(I i , X) (i = 1, . . . , n) are Banach spaces.
Abstract formulation
Let us reformulate problem (5)-(8) [(5)-(7) and (8’), respectively] as the problem
of solving the abstract equation
Lu = {f, ϕ1 , . . . , ϕn } ,
where L is the operator given by the formula
Lu := {Lu, `1 u, . . . , `n u} .
6
Nonlocal pluriparabolic problems
EJDE–2001/21
We consider L as an unbounded operator with the domain D(L) consisting of
∂u
1
1
∈ L2 (I, B2,σ
(Ω))
all functions u belonging to L2 (I, B2,σ
(Ω)), such that ∂t
i
2
∂ u
2
1
(i = 1, . . . , n), ∂u
∂x , ∂x2 ∈ L (I, B2,σ (Ω)) and such that u satisfies conditions (7)
and (8) [(7) and (8’), respectively].
Let B be the Hilbert space obtained by completing of the domain D(L) in
the norm
kukB =
n
2
kP (t)ukL2 (I,B 1
2,σ
(Ω)) +
n
X
2
kukC(I i ,L2 (∆i ,L2 (Ω)))
σ
o1/2
.
i=1
By completing of the set D(L) with respect to the norm
kukB =
n
2
kukL2 (I,H 1 (Ω)) +
n
X
2
kukC(I i ,L2 (∆i ,L2 (Ω)))
σ
o1/2
,
i=1
we obtain a Banach space B 0 . The elements u ∈ B [B 0 , respectively] are continuous functions on Ii (i = 1, . . . , n) with the values in L2 (∆i , L2σ (Ω)) (i = 1, . . . , n).
Hence, the mappings
`i : B 3 u → `i u = u|ti =0 ∈ L2 (∆i , L2σ (Ω)) (i = 1, . . . , n),
are defined and continuous on B [B 0 , respectively] .
Qn
1
We denote by F the Hilbert space L2 (I, B2,σ
(Ω))× i=1 L2 (∆i , L2σ (Ω)). The
1
elements of F are of the form {f, ϕ1 , . . . , ϕn }, where f ∈ L2 (I, B2,σ
(Ω)), ϕi ∈
2
2
L (∆i , Lσ (Ω)) (i = 1, . . . , n), and the norm is defined by:
k{f, ϕ1 , . . . , ϕn }kF =
n
2
kf kL2 (I,B 1
2,σ
(Ω)) +
n
X
2
kϕi kL2 (∆i ,L2σ (Ω))
o1/2
.
i=1
If the operator L is closable then we denote by L the closure of L and by
D(L) its domain.
Definition 3.5 A solution of the abstract equation
Lu = {f, ϕ1 , . . . , ϕn }
is called a strongly generalized solution of problem (5)-(8) [(5)-(7) and (8’),
respectively].
In concluding this section, we shall state the following lemma, which can be
applied to obtain a priori estimates for the solutions.
Lemma 3.1 If fk (k = 1, 2, 3) are nonnegative functions on I; f1 and f2 are
integrable on I; τ → f3 (τ ) is non-decreasing with respect to τi (i = 1, . . . , n)
and
Z
n Z τi
X
f1 (t)dt + f2 (τ ) ≤ c
f2 (τ1 , . . . , τi−1 , ti , τi+1 , . . . , τn ) + f3 (τ ),
Iτ
i=1
0
EJDE–2001/21
Abdelfatah Bouziani
7
where c is a positive constant, then
(
Z
f1 (t)dt + f2 (τ ) ≤ exp nc
Iτ
n
X
τi
)
· f3 (τ ),
i=1
Qn
where I τ := i=1 (0, τi ), 0 < τi < Ti (i = 1, . . . , n).
The proof of this lemma is analogous to the proof of Lemma 1 in [8].
4
Uniqueness and continuous dependence
In this section we will establish a priori estimates. Thus we will deduce the
uniqueness and continuous dependence of the solutions upon the data.
Theorem 4.1 If Assumptions A1 and A2 are satisfied then there is a constant
c > 0, independent of u such that
kukB ≤ c kLukF
(10)
[kukB 0 ≤ c kLukF 0 , respectively]
(100 )
for u ∈ D(L).
1
Proof. Taking the square of the norm in the space L2 (I τ , B2,σ
(Ω)) of Lu, we
get
Z
Iτ
Z
n
X
∂u 2
2
dt +
kP (t)ukB 1 (Ω) dt
1
2
∂t
B2,σ (Ω)
τ
i
I
i=1
Z
n Z
X
∂u
2
−2
=x P (t)u · =x (σ
)dx =
kf kB 1 (Ω) dt ;
2,σ
∂t
τ
τ
i
I
i=1 Ω×I
respectively
Z
Iτ
n
X ∂u 2
∂ti 1
i=1
dt +
Z
B2,σ (Ω)
−2
n Z
X
i=1
p2 (Jx σ, t)(
Ω×I τ
Ω×I τ
∂u 2
) dx
∂x
∂u
∂u
p(Jx σ, t)
· =x (σ
)dx =
∂x
∂ti
Z
Iτ
Observe that
−2
n Z
X
=x P (t)u · =x (σ
Ω×I τ
"
i=1
n Z
X
−2
i=1
Ω×I τ
∂u
)dx
∂ti
#
∂u
∂u
p(Jx σ, t)
· =x (σ
)dx
∂x
∂ti
2
kf kB 1
2,σ (Ω)
dt .
8
Nonlocal pluriparabolic problems
n Z
X
=
σ(x)p(Jx σ, ti,τ )u2 (x, ti,τ )dxdti
Qi
i=1
−
EJDE–2001/21
n Z
X
σ(x)
τ
i=1 Ω×I
Z
n
X
+2
i=1
∂p(Jx σ, t) 2
u dxdt
∂ti
σ(x)
Ω×I τ
∂u
∂p(Jx σ, t)
u · =x (σ
)dxdt,
∂Jx σ
∂ti
it yields
2
n
X
∂u ∂ti 1
Z
Iτ
i=1
n Z
X
i=1
Iτ
2
Iτ
B2,σ (Ω)
kP (t)ukB 1 (Ω) dt
(11)
2
σ(x)p(Jx σ, ti,τ )u2 (x, ti,τ )dxdti
n
X ∂u 2
∂ti 1
i=1
n Z
X
dt +
Z
B2,σ (Ω)
σ(x)p(Jx σ, ti,τ )u2 (x, ti,τ )dxdti
2
kf kB 1
2,σ
Iτ
(Ω) dt +
n Z
X
i=1
n Z
X
σ(x)
Ω×I τ
i=1
p2 (Jx σ, t)(
Ω×I τ
Qi
i=1
Z
+2
Z
Qi
hZ
=
dt +
n Z
X
i=1
∂u 2
) dx
∂x
i
σ(x)p(Jx σ, ti,0 )ϕ2 (x, ti )dxdti
Qi
∂p(Jx σ, t) 2
u dxdt
∂ti
σ(x)
Ω×I τ
∂p(Jx σ, t)
∂u
u · =x (σ
)dxdt.
∂Jx σ
∂ti
In light of the Cauchy inequality, the last sum of integrals on the right-hand
side of (11) are dominated as follows
n Z
X
2
i=1
≤
Z
Ω×I τ
∂p(Jx σ, t)
∂u
u · =x (σ
)dxdt
∂Jx σ
∂ti
Ω×I τ
Z X
n
∂p(Jx σ, t) 2 2
∂u 2
σ 2 (x)(
) u dxdt +
dt,
1
∂Jx σ
∂ti B2,σ
(Ω)
Iτ
i=1
σ(x)
where the second term will be absorbed in the left-hand side of (11).
By virtue of Assumptions A1-A2 and the Poincarré inequality [26]:
Z
a
b
(b − a)2
u (x, .)dx ≤
2
2
Z
a
b
∂u(x, .) 2
1
(
) dx +
∂x
b−a
(Z
a
b
)2
u(x, .)dx
EJDE–2001/21
Abdelfatah Bouziani
we obtain
Z
n Z
X
2
kP (t)ukB 1 (Ω) dt +
2
Iτ
i=1
≤ c5
(Z
Iτ
2
kf kB 1 (Ω)
2,σ
dt +
≤
∆τi
i=1
c05
(Z
Iτ
u(., ti,τ )2 2
dti
L (Ω)
2
kf kB 1 (Ω)
2,σ
(12)
σ
n Z
X
∆τi
i=1
respectively
Z
n Z
X
2
kukH 1 (Ω) dt +
Iτ
∆τi
9
ϕi (., ti )2 2
dti
L (Ω)
)
σ
+ c6
Z
2
Iτ
kukL2σ (Ω) dt
u(., ti,τ )2 2
dti
L (Ω)
(12’)
σ
dt +
n Z
X
∆τi
i=1
ϕi (., ti )2 2
dti
L (Ω)
)
σ
+ c06
Z
Iτ
2
kukL2σ (Ω) dt,
where
max(1, c1 )
c5 :=
min(1, c0 )
"
c4 c23 + nc2
c6 :=
min(1, c0 )
"
c05
max(1, c1 )
:=
c20
c0 2
2 , ( b−a ) )
min(c0 ,
c06 :=
, respectively ,
c4 c23 + nc2
min(c0 ,
#
c20
c0 2
2 , ( b−a ) )
, respectively
#
and
∆τi = (0, τ1 ) × . . . × (0, τi−1 ) × (0, τi+1 ) × . . . × (0, τn ) (i = 1, . . . , n).
To eliminate the last term on the right-hand side of the above inequality, we
apply Lemma 3.1 by denoting the first integral of the left-hand side by f1 (τ ),
the sum of the last integrals on the same side by f2 (τ ), and the first integral
and the sum of integrals on ∆τi by f3 (τ ). Consequently, we get
2
kP (t)ukL2 (I τ ,B 1 (Ω)) +
2
n
X
2
ku(. . . , τi , . . .)kL2 (∆τ ,L2σ (Ω))
i
(13)
i=1
n
2
≤ c7 kf kL2 (I,B 1
2,σ
(Ω)) +
n
X
2
kϕi kL2 (∆i ,L2σ (Ω))
o
;
i=1
respectively
2
kukL2 (I τ ,H 1 (Ω)) +
n
X
2
ku(. . . , τi , . . .)kL2 (∆τ ,L2σ (Ω))
i
(13’)
i=1
n
2
≤ c07 kf kL2 (I,B 1
2,σ
(Ω)) +
n
X
2
kϕi kL2 (∆i ,L2σ (Ω))
o
i=1
where c7 := c5 exp nc6
Pn
i=1
Pn
Ti [c07 := c05 exp nc06 i=1 Ti , respectively].
10
Nonlocal pluriparabolic problems
EJDE–2001/21
The right-hand side of (13) [(13’), respectively] is independent of τi (i =
1, . . . , n). Hence replacing the left-hand side by its upper bound with respect
to τi from 0 to Ti (i = 1, . . . , n), we obtain (10) [(10’), respectively] with c =
1
0
1
c72 [c = c7 2 , respectively]. The proof of Theorem 4.1 is complete.
♦
Remark 4.1 We can obtain such estimate for a special case of problem (1)(3) and (4’), for which P (t) is the Bessel operator, i.e., for which σ(x) :=
x, p(Jx σ, t) := x, Ω := (0, b), if we take the inner product in L2 (Ω × I) of
the
equation
the following integro-differential operator: M u =
Pconsidered
Pn with
n
∂u
∂u
2
+
x
−
x ∂u
♦
x i=1 =x ∂t
i=1 ∂ti
∂x .
i
It follows from estimation (10) [(10’), respectively] that there is a bounded
inverse operator L−1 on the range R(L) of L. However, since we have no
information concerning R(L), except that R(L) ⊂ F , we must extend L so that
the estimation (10) [(100 ), respectively] holds for the extension and its range
is the whole space. We first show that L : B [B 0 , respectively]→ F , with the
domain D(L), has a closure, i.e., the closure of the graph Γ(L) ⊂ B × F [B 0 × F ,
respectively] of L is the graph Γ(L) = Γ(L) of the new linear operator L.
Proposition 4.1 If the assumptions of Theorem 4.1 are satisfied, then the operator L with domain D(L) is closable.
Proof. The proof of this proposition is similar to the proof of Proposition 1 in
Bouziani [8].
♦
The following corollaries are immediate consequences of Theorem 4.1 and of
Proposition 4.1.
Corollary 4.1 Suppose that the assumptions of Theorem 4.1 are satisfied. Then
there is a constant c > 0 such that
kukB ≤ c LuF for u ∈ D(L)
(14)
respectively
kukB 0 ≤ c LuF 0
for u ∈ D(L).
(140 )
Corollary 4.2 If the assumptions of Theorem 4.1 are fulfilled and problem (5)(8) [(5)-(7) and (8’) respectively] has a strongly generalized solution then this
solution is unique and depends continuously on {f, ϕ1 , . . . , ϕn }.
Corollary 4.3 If the assumptions of Theorem 4.1 are satisfied then R(L) =
−1
R(L) and L = L−1 , where R(L) and R(L) denote the set of values taken by
L and L, respectively, and L−1 is the continuous extension of L−1 from R(L)
to R(L).
5
Existence of a solution
We are, now, in a position to state and to prove the main result of this paper:
EJDE–2001/21
Abdelfatah Bouziani
11
Theorem 5.1 If the assumptions of Theorem 4.1 are fulfilled then, for any
1
f ∈ L2 (I, B2,σ
(Ω)) and ϕi ∈ L2 (∆i , L2σ (Ω)) (i = 1, . . . , n), problem (5)-(8) [(5)(7) and (8’), respectively] admits a unique strongly generalized solution u =
−1
L {f, ϕ1 , . . . , ϕn } such that
u∈
n
Y
C(I i , L2 (∆i , L2σ (Ω))),
P (t) ∈ L2 (I, B21 (Ω))
i=1
"
2
1
u ∈ L (I, H (Ω)) ∩
n
Y
2
C(I i , L
(∆i , L2σ (Ω))),
#
respectively .
i=1
Proof. Corollary 4.3 states that to prove the existence of the solution, in the
sense of Definition 3.5, for any {f, ϕ1 , . . . , ϕn }, it is sufficient to prove that
R(L) does not have an orthogonal complement in F . For this purpose we need:
1
Proposition 5.1 If, for some function ω ∈ L2 (I, B2,σ
(Ω)) and for all u ∈ D(L)
with `i u = 0 (i = 1, . . . , n), we have
1 (Ω)) = 0
(Lu, ω)L2 (I,B2,σ
(15)
then ω ≡ 0 almost everywhere in Ω × I.
Assume for the moment that Proposition 5.1 has been proved and return to
the proof of Theorem 5.1.
Let W = (ω, ω1 , . . . , ωn ) ∈ F be orthogonal to the set R(L). Consequently,
1 (Ω)) +
(Lu, ω)L2 (I,B2,σ
n
X
(`i u, ωi )L2 (∆i ,L2σ (Ω)) = 0,
∀u ∈ D(L).
(16)
i=1
Assume in (16) that u is any element of D(L) such that `i u = 0 (i = 1, . . . , n).
We then conclude, by Proposition 5.1, that ω ≡ 0 almost everywhere in Ω × I.
Thus, (16) implies that
n
X
(`i u, ωi )L2 (∆i ,L2σ (Ω)) = 0,
∀u ∈ D(L).
i=1
Since `i u (i = 1, . . . , n) are independent and the ranges of the operators `i (i =
1, . . . , n) are dense in L2 (∆i , L2σ (Ω))(i = 1, . . . , n) , it follows that ωi ≡ 0 (i =
1, . . . , n) almost everywhere in Ω × ∆i (i = 1, . . . , n).
♦
To complete the proof of Theorem 5.1, it remains to prove Proposition 5.1.
Proof of Proposition 5.1. Equality (15) can be written in the form
n Z
X
i=1
∂u
1 (Ω) dt =
(
, ω)B2,σ
∂t
i
I
Z
I
(P (t)u, σω)B21 (Ω) dt.
12
Nonlocal pluriparabolic problems
EJDE–2001/21
In the above formula, we put
u=
n
X
n Z
X
=ti (ekτi θ) =
i=1
i=1
ti
ekτi θdτi ,
0
where k is a constant such that
k≥
1
(c2 + c23 c4 ),
c0
(17)
θ satisfies conditions
1
θ ∈ L2 (I, B2,σ
(Ω)),
n
X
1
=ti (ekτi θ) ∈ L2 (I, B2,σ
(Ω)),
i=1
p(Jx σ, t)(
∂=ti (ekτi θ)
) ∈ L2 (I, L2 (Ω)),
∂x
P (t)=ti (ekτi θ) ∈ L2 (I, B21 (Ω)),
and u verifies conditions (7), (8) [(7), (8’), respectively]. Consequently, we get
n Z
X
i=1
I
1 (Ω) dt =
ekti (θ, ω)B2,σ
n Z
X
i=1
I
(P (t)=ti (ekτi θ), σω)B21 (Ω) dt
(18)
The left-hand side of (18) shows that the mapping
1
L2 (I, B2,σ
(Ω)) 3 θ →
n Z
X
i=1
I
(P (t)=ti (ekτi θ), σω)B21 (Ω) dt
is a linear continuous functional if the function ω, on the right-hand side of (22),
satisfies the following properties:
1
ω ∈ L2 (I, B2,σ
(Ω)),
(19)
=2x (σω) ∈ L2 (I, L2 (Ω)),
(20)
∂=∗ti (p(Jx σ, t)=x (σω))
∈ L2 (I, L2 (Ω)),
∂x
∂ ekti ∂=∗ti (p(Jx σ, t)=x (σω))
(
) ∈ L2 (I, L2 (Ω)),
∂x σ(x)
∂x
(21)
=2x (σω)|x=b = 0,
(23)
=x (σω)|x=b = 0,
(24)
(22)
and
where
=∗ti η =
Z
Ti
ti
η(Jx σ, ti,τ )dτi (i = 1, . . . , n)
EJDE–2001/21
Abdelfatah Bouziani
13
[Here properties (20) and (23) are specific to problem (5)-(8)].
By replacing ω by θ in equality (18), the standard integration by parts of
the right-hand side of the obtained equality leads to
n Z
X
2
ekti kθkB 1 (Ω) dt
(25)
i=1
2,σ
I
n
= −
1X
2 i=1
n
−
−
1X
2 i=1
n Z
X
Z
σ(x)e−kTi p(Jx σ, ti,T )(=Ti (ekti θ))2 dxdti
Ω×∆i
Z
σ(x)e−kti (kp(Jx σ, t) −
Ω×I
σ(x)
Ω×I
i=1
∂p(Jx σ, t)
)(=ti (ekτi θ))2 dxdt
∂ti
∂p(Jx σ, t)
=ti (ekτi θ)=x (σθ)dxdt.
∂Jx σ
Applying the Cauchy inequality to the last integral on the right-hand side of
(25), we obtain
n Z
X
2
ekti kθkB 1 (Ω) dt
i=1
2,σ
I
= −
1
2
n Z
X
i=1
n
1X
−
2 i=1
σ(x)e−kTi p(Jx σ, ti,T )(=Ti (ekti θ))2 dxdti
Ω×∆i
Z
−kti
σ(x)e
Ω×I
∂p
∂p 2
kp −
− 2σ(x)(
) (=ti (ekτi θ))2 dxdt.
∂ti
∂Jx σ
Since the integrals on Ω × ∆i (i, . . . , n) are negative then it follows, by Assumptions A1-A2, that
n Z
X
2
ekti kθkB 1 (Ω) dt
i=1
≤
2,σ
I
−(kc0 − c2 −
2c23 c4 )
n Z
X
i=1
σ(x)e−kti (=ti (ekτi θ))2 dx dt.
Ω×I
The above inequality and (17) imply that
n Z
X
2
ekti kθkB 1
i=1
I
2,σ (Ω)
dt ≤ 0,
and thus ω ≡ 0 almost everywhere in Ω × I. The proof of Theorem 5.1 is
complete.
♦
Remark 5.1 Applying a similar argumentation to those given above, our results
can be generalized to the following nonlocal pluriparabolic problem:
(Lv)(x, t) :=
n
n
X
Y
∂v
1
2
−
sign (1 − |λi | )P (t)v = g(x, t)
∂t
σ(x)
i
i=1
i=1
14
Nonlocal pluriparabolic problems
EJDE–2001/21
for (x, t) ∈ Ω × I,
(`i v)(x, t) := v(x, ti,0 ) − λi v(x, ti,T ) = Φi (x, ti )
for (x, t) ∈ Qi,0 (i = 1, . . . , n),
Z
σ(x)v(x, t)dx = E(t) for t ∈ I,
ZΩ
xσ(x)v(x, t)dx = G(t) for t ∈ I,
Ω
∂v(a, t)
= H(t) for t ∈ I, respectively .
∂x
References
[1] N. E. Benouar and N. I. Yurchuk, Mixed problem with an integral condition for parabolic equations with the Bessel operator, Differentsial’nye
Uravneniya, 27 (1991), 2094-2098.
[2] A. Bouziani, On a class of parabolic equations with a nonlocal boundary
condition, Acad. Roy. Belg. Bull. Cl. Sci., 10 (1999), 61-77.
[3] A. Bouziani, Solution forte d’un problème mixte avec une condition non
locale pour une classe d’équations paraboliques, Maghreb Math. Rev., 6
(1997), 1-17.
[4] A. Bouziani, Mixed problem with boundary integral conditions for a certain
parabolic equation, J. Appl. Math. Stochastic Anal., 9 (1996), 323-330.
[5] A. Bouziani, Solution forte d’un problème mixte avec une condition non
locale pour une classe d’équations hyperboliques, Acad. Roy. Belg. Bull.
Cl. Sci., 8 (1997), 53-70.
[6] A. Bouziani, On a third order parabolic equation with a nonlocal boundary
condition, J. Appl. Math. Stochastic. Anal., 13 (2000), 181-195.
[7] A. Bouziani, On a class of nonclassical hyperbolic equations with nonlocal
conditions, J. Appl. Math. Stochastic. Anal., (to appear).
[8] A. Bouziani, Strong solution for a mixed problem with nonlocal condition
for a certain pluriparabolic equations, Hiroshima Math. J., 27 (1997), 373390.
[9] A. Bouziani and N. E. Benouar, Problème mixte avec conditions intégrales
pour une classe d’équations paraboliques, C.R.A.S, Paris, 321, Série I,
(1995), 1177-1182.
[10] A. Bouziani and N. E. Benouar, Mixed problem with integral conditions
for a third order parabolic equation, Kobe J. Math., 15 (1998), 47-58.
EJDE–2001/21
Abdelfatah Bouziani
15
[11] L. Byszewski, Existence and uniqueness of solutions of nonlocal problems
for hyperbolic equation uxt = F (x, t, u, ux ), J. Appl. Math. Stochastic.
Anal., 3 (1990), 163-168.
[12] L. Byszewski, Theorem about existence and uniqueness of continuous solutions for nonlocal problem for nonlinear hyperbolic equation, Applicable
Analysis, 40 (1991), 173-180.
[13] L. Byszewski, Differential and functional-differential problems together
with nonlocal conditions, Cracow University of Technology Monograph,
184, Cracow 1995.
[14] L. Byszewski and V. Lakshmikantham, Monotone iterative technique for
nonlocal hyperbolic differential problem, J. Math. Phys. Sci., 26 (1992),
345-359
[15] L. Byszewski and N. S. Papageorgiou, An application of a noncompactness
technique to an investigation of the existence of solutions to a nonlocal
multivalued Darboux problem, J. Appl. Math. Stochastic. Anal., 12 (1999),
179-190.
[16] J. R. Cannon, The one-dimensional heat equation, in: Encyclopedia of
mathematics and its applications, 23 , Addison-Wesley, Reading, MA,
Menlo Park, CA, 1984.
[17] J. R. Cannon, S. P. Esteva and J. van der Hoek, A Galerkin procedure
for the diffusion equation subject to the specification of energy, SIAM J.
Numer. Anal., 24 (1987), 499-515.
[18] J. R. Cannon and J. van der Hoek, The existence and the continuous dependence for the solution 0f the heat equation subject to the specification
of energy, Boll. Uni. Math. Ital. Suppl., 1 (1981), 253-282.
[19] J. R. Cannon and J. van der Hoek, An implicit finite difference scheme
for the diffusion of mass in a portion of the domain, Numerical Solutions
of Partial Differential Equations (J. Noye, Ed), 527-539, North-Holland,
Amsterdam, 1982.
[20] N. I. Ionkin, Solution of boundary value problem in heat conduction theory with nonlocal boundary conditions, Differentsial’nye Uravneniya, 13
(1977), 294-304.
[21] N. I. Kamynin, A boundary value problem in the theory of the heat conduction with non classical boundary condition, Th. Vychisl. Mat. Mat., Fiz.,
43 (1964), 1006-1024.
[22] S. Mesloub and A. Bouziani, Problème mixte avec conditions aux limites intégrales pour une classe d’équations paraboliques bidimensionnelles,
Acad. Roy. Belg. Bull. Cl. Sci., 9 (1998), 61-72.
16
Nonlocal pluriparabolic problems
EJDE–2001/21
[23] S. Mesloub and A. Bouziani, On a class of singular hyperbolic equation with
a weighted integral condition, Intern. J. Math. & Math. Sci., 22 (1999),
511-520.
[24] L. S. Pulkina, A non-local problem with integral conditions for hyperbolic
equations, Electronic Journal of Differential Equations, 1999 (1999), No
45, 1-6.
[25] L. S. Pulkina, On the solvability in L2 of a nonlocal problem with integral
conditions for an hyperbolic equation, (2000), No 2, 1-6.
[26] K. Rektorys, Variational methods in mathematics, science, and engineering, Reidel Publishing Company, Dordrecht-Boston-London, 1982.
[27] N. I. Yurchuk, Mixed problem with an integral condition for certain
parabolic equations, Differentsial’nye Uravneniya, 22 (1986), 2117-2126.
[28] N. I. Yurchuk, Problème avec conditions aux limites non locales pour les
équations opérationnelles hyperboliques avec des dérivées partielles du second ordre, Differentsial’nye Uravneniya, 11 (1975), 1687-1693.
Abdelfatah Bouziani
Département de Mathématiques,
Centre Universitaire Larbi Ben M’hidi-Oum El Bouagui,
B.P. 565, 04000, Algérie.
e-mail: af bouziani@hotmail.com