Hindawi Publishing Corporation
Journal of Applied Mathematics and Stochastic Analysis
Volume 2009, Article ID 975601, 13 pages
doi:10.1155/2009/975601
Research Article
A Boundary Value Problem with Multivariables
Integral Type Condition for Parabolic Equations
A. L. Marhoune1 and F. Lakhal2
1
Laboratory Equations Différentielles, Departement of Mathematics, University Mentouri Constantine,
Constantine 25017, Algeria
2
Department of Mathematics, Science University of 08 Mai 45, P.O. Box 401, Guelma 24000, Algeria
Correspondence should be addressed to A. L. Marhoune, ahmed marhoune@hotmail.com
Received 27 January 2009; Revised 14 May 2009; Accepted 13 October 2009
Recommended by Sergiu Aizicovici
We study a boundary value problem with multivariables integral type condition for a class of
parabolic equations. We prove the existence, uniqueness, and continuous dependence of the
solution upon the data in the functional wieghted Sobolev spaces. Results are obtained by using a
functional analysis method based on two-sided a priori estimates and on the density of the range
of the linear operator generated by the considered problem.
Copyright q 2009 A. L. Marhoune and F. Lakhal. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
Certain problems of modern physics and technology can be effectively described in terms of
nonlocal problems with integral conditions for partial differential equations. These nonlocal
conditions arise mainly when the data on the boundary cannot be measured directly.
Motivated by this, we consider in the rectangular domain Ω 0, 1 × 0, T , the following
nonclassical boundary value problem of finding a solution ux, t such that
Lu ∂2 u
∂u
− at 2 fx, t,
∂t
∂x
1.1
where the function at and its derivative are bounded on the interval 0, T :
0 < c0 ≤ at ≤ c1 ,
0 < c2 ≤
dat
≤ c3,
dt
1.2
2
Journal of Applied Mathematics and Stochastic Analysis
lu ux, 0 ϕx,
x ∈ 0, 1,
u0, t u β, t u γ, t u1, t,
α
1.3
t ∈ 0, T ,
1.4
γ
1
ux, tdx 2 ux, tdx ux, tdx 0,
0
β
1.5
δ
α 1 − δ γ − β, t ∈ 0, T .
0 < α < β < γ < δ < 1,
Here, we assume that the known function ϕ satisfies the conditions given in 1.4 and 1.5,
that is,
ϕ0 ϕ β ϕ γ ϕ1,
α
γ
1
ϕxdx 2 ϕxdx ϕxdx 0.
0
β
1.6
δ
When considering the classical solution of the problem 1.1–1.5, along with 1.5, there
should be the fulfilled conditions:
a0 ϕ 0 ϕ β − ϕ γ − ϕ 1 f1, 0 f γ, 0 − f β, 0 − f0, 0,
α
α
γ
1
γ
1
ϕ xdx 2
a0
0
ϕ xdx β
ϕ xdx
δ
fx, 0dx 2
0
fx, 0dx β
fx, 0dx.
δ
1.7
Mathematical modelling of different phenomena leads to problems with nonlocal or
integral boundary conditions. Such a condition occurs in the case when one measures an
averaged value of some parameter inside the domain. This amounts to the specification of
the energy or mass contained in a portion of the conductor or porous medium as a function
of time. This problems arise in plasma physics, heat conduction, biology and demography, as
well as modelling of technological process, see, for example, 1–5. Boundary-value problems
for parabolic equations with integral boundary condition are investigated by Batten 6,
Bouziani and Benouar 7, Cannon 8, 9, Cannon, Perez Esteva and van der Hoek 10,
Ionkin 11, Kamynin 12, Shi and Shillor 13, Shi 4, Marhoune and Bouzit 14, Marhoune
and Hameida 15, Yurchuk 16, and many references therein. The problem with onevariable resp., two-variable boundary integral type condition is studied in 5 and by
Marhoune and Latrous 17 resp., in Marhoune 2.
Mention that in the cited paper 16, the author proved the existence, uniqueness, and
continuous dependence of a stronge solution in weighted Sobolev spaces to the problem
∂u
∂
∂u
ax, t
fx, t,
∂t
∂x
∂x
1.8
Journal of Applied Mathematics and Stochastic Analysis
3
under the following conditions:
ux, 0 0,
0 ≤ x ≤ 1,
u0, t 0, 0 < t ≤ T,
1
ux, tdx 0.
1.9
0
This last integral condition in the form
1
ux, tdx mt,
1.10
0 < t ≤ T,
0
arises, for example, in biochemistry in which m is a constant, and in this case is known as the
conservation of protein 18. Further, in 5, the author studied a similar problem with the
weak integral condition
α
ux, tdx 0,
1.11
0 < α < 1.
0
The same problem with the new integral condition
α
ux, tdx 0
1
ux, tdx 0,
α β 1,
1.12
β
was investigated in 2. The present paper is an extension in the same direction. By
constructing a suitable multiplicator, we will try to establish existence and uniqueness of
solution of problem 1.1–1.5. Note that the multivariables integral type condition 1.5 is
considerably much weaker and better than that used in 2. In fact, some physical problems
have motivated specialists to consider nonlocal integral condition 1.5, which tells us the
integral total effect of the solution u over several independent portions 0, α, β, γ, and δ, 1
of interval I 0, 1 at certain time t that give this effect over the entire or part of this interval.
We associate with 1.1–1.5 the operator L L, l, defined from E into F, where E is
the Banach space of functions u ∈ L2 Ω, satisfying 1.4 and 1.5, with the finite norm
⎤
⎡
1
2 2 2
px
px ∂u 2
⎣ ∂u ∂ u ⎦dxdt sup
u2E ∂t ∂x2 ∂x dx
Ω 3
0≤t≤T 0 2
sup
0≤t≤T
α
0
|u|2 dx sup
0≤t≤T
γ
β
|u|2 dx sup
0≤t≤T
1
δ
|u|2 dx,
1.13
4
Journal of Applied Mathematics and Stochastic Analysis
and F is the Hilbert space of vector-valued functions F f, ϕ obtained by completion of
the space L2 Ω × W22 0, 1 with respect to the norm
F2F
2
f, ϕF px 2
f dxdt Ω 3
1
α
γ
1
2
2
2
px dϕ 2
ϕ dx ϕ dx ϕ dx,
dx dx 2
0
0
β
δ
1.14
where
⎧
2
⎪
⎪
⎪x ,
⎪
⎪
⎪
⎪
⎨ γ − β 2,
px 2
2 ⎪
⎪
γ −x x−β ,
⎪
⎪
⎪
⎪
⎪
⎩1 − x2 ,
x ∈ 0, α,
x ∈ α, β ∪ γ, δ ,
x ∈ β, γ ,
1.15
x ∈ δ, 1.
Using the energy inequalities method proposed in 16, we establish two-sided a priori
estimates. Then, we prove that the operator L is a linear homeomorphism between the spaces
E and F.
2. Two-Sided A Priori Estimates
Theorem 2.1. For any function u ∈ E, one has the a priori estimate
2.1
Lu2F ≤ c4 u2E ,
where the constant c4 is independent of u. In fact, c4 2 max1, c12 .
Proof. Using 1.1 and initial condition 1.3, we obtain
⎡
⎤
2
∂2 u 2
px
px
∂u
⎦
⎣ c12 |Lu|2 dxdt ≤ 2
dxdt,
∂t 2
3
3
∂x Ω
Ω
1
1
px dϕ 2
px ∂u 2
dx dx ≤ sup
∂x dx,
0 2
0≤t≤T 0 2
γ
β
2
ϕ dx ≤ sup
0≤t≤T
γ
β
2
|u| dx,
1
δ
α
2
ϕ dx ≤ sup
0
0≤t≤T
2
ϕ dx ≤ sup
0≤t≤T
Combining the inequalities in 2.2, we obtain 2.1 for u ∈ E.
1
δ
α
0
|u|2 dx.
|u|2 dx,
2.2
Journal of Applied Mathematics and Stochastic Analysis
5
Theorem 2.2. For any function u ∈ E, one has the a priori estimate
u2E ≤ c5 Lu2F ,
2.3
expcT max49, 2c1 ,
min 13/32, c0 , c02 /2
2.4
with the constant
c5 and c is such that
cc0 ≥ c3 .
2.5
Before proving this theorem, we need the following lemma.
Lemma 2.3 see 19. For u ∈ E, one has
2
b
b b
2
∂u ∂uξ, t dξ dx ≤ 4 x − a2 dx,
∂t
∂t
a x
a
b
b x
2
∂uξ, t 2
2 ∂u dξ dx ≤ 4 b − x dx.
∂t
∂t
a
a
a
2.6
Proof of Theorem 2.2. Define
⎧
α
∂uξ, t
⎪
2 ∂u
⎪
2x
dξ,
x
⎪
⎪
⎪
∂t
∂t
⎪
x
⎪
⎪
2 ∂u
⎪
⎪
⎪
γ −β
,
⎪
⎪
∂t
⎪
⎪
⎨
2 ∂u
x ∂uξ, t
2 ∂u x−β
2 γ −x
dξ
γ −x
Mu ∂t
∂t
∂t
⎪
β
⎪
γ
⎪
⎪
∂uξ, t
⎪
⎪
⎪
2 x − β
dξ,
⎪
⎪
∂t
⎪
x
⎪
⎪
x
⎪
∂u
∂uξ, t
⎪
⎪
21 − x
dξ,
⎩1 − x2
∂t
∂t
δ
x ∈ 0, α,
x ∈ α, β ∪ γ, δ ,
2.7
x ∈ β, γ ,
x ∈ δ, 1.
We consider for u ∈ E the quadratic formula
τ1
Re
exp−ctLuMu dxdt,
0
0
2.8
6
Journal of Applied Mathematics and Stochastic Analysis
with the constant c satisfying 2.5, obtained by multiplying 1.1 by exp−ctMu, by
integrating over Ωτ , where Ωτ 0, 1 × 0, τ, with 0 ≤ τ ≤ T , and by taking the real part.
Integrating by parts in 2.8 by report to x with the use of boundary conditions 1.4 and
1.5, we obtain
τ1
exp−ctLuMudxdt
Re
0
0
τα
2
α
∂u ∂uξ, t 2
dξ dxdt
px exp−ct dxdt exp−ct
∂t
∂t
0
0 0
x
τ1
0
0
⎡
⎤
x ∂uξ, t 2 γ ∂uξ, t 2
dξ dξ ⎦dxdt
exp−ct⎣
β ∂t
∂t
β
x
τγ
τ1
0
δ
2 Re
τ1
x
∂uξ, t 2
∂u ∂2 u
dξ dxdt Re
dxdt
exp−ct
exp−ctpxa
∂t
∂x ∂x∂t
δ
τα
exp−ctau
0
2 Re
0
τ1
exp−ctau
0
δ
0
∂u
dxdt 4 Re
∂t
0
τγ
exp−ctau
0
β
∂u
dxdt
∂t
∂u
dxdt.
∂t
2.9
On the other hand, by using the elementary inequalities we get
τ1
Re
exp−ctLuMudxdt
0
≥
0
τ1
0
τ1
2
∂u ∂u ∂2 u
dxdt
px exp−ct dxdt Re
exp−ctpxa
∂t
∂x ∂x∂t
0
0 0
2 Re
τα
0
2 Re
∂u
exp−ctau dxdt 4 Re
∂t
0
τ1
exp−ctau
0
δ
∂u
dxdt.
∂t
τγ
exp−ctau
0
β
∂u
dxdt
∂t
2.10
Journal of Applied Mathematics and Stochastic Analysis
7
Again, integrating by parts the second, third, fourth, and fifth terms of the right-hand
side of the inequality 2.10 by report to t and taking into account the initial condition
1.3 and 2.5 gives
τ1
exp−ctpxa
Re
0
0
∂u ∂2 u
dxdt ≥
∂x ∂x∂t
1
∂ux, τ 2
exp−cτ
dx
pxaτ
2
∂x 0
1
2
−
τα
Re
exp−ctau
0
0
∂u
dxdt ≥
∂t
τγ
Re
0
∂u
exp−ctau dxdt ≥
∂t
β
τ1
Re
exp−ctau
0
δ
∂u
dxdt ≥
∂t
1
dlu 2
dx,
pxa0
dx 0
α
α
γ
γ
1
1
exp−cτ
aτ|ux, τ|2 dx −
2
0
exp−cτ
aτ|ux, τ|2 dx −
2
β
exp−cτ
aτ|ux, τ|2 dx −
2
δ
a0
|lu|2 dx;
0 2
a0
|lu|2 dx,
β 2
a0
|lu|2 dx.
2
δ
2.11
Using 2.11 in 2.10, we get
τ1
Re
0
exp−ctLuMudxdt 0
γ
a0|lu| dx 2 a0|lu|2 dx
2
0
1
a0|lu|2 dx δ
≥
α
1
2
β
1
dlu 2
dx
pxa0
dx 0
τ1
0
1
2
2
∂u px exp−ct dxdt
∂t
0
1
α
∂ux, τ 2
exp−cτpxaτ
dx exp−cτaτ|ux, τ|2 dx
∂x 0
0
γ
1
2
2 exp−cτaτ|ux, τ| dx exp−cτaτ|ux, τ|2 dx.
β
δ
2.12
8
Journal of Applied Mathematics and Stochastic Analysis
By using the ε-inequalities on the first integral in the left-hand side of 2.12 and Lemma 2.3,
we obtain
15
32
τ1
0
2
∂u ∂ux, τ 2
1 1
dx
px exp−ct dxdt exp−cτpxaτ
∂t
2 0
∂x 0
α
γ
2
exp−cτaτ|ux, τ| dx 2 exp−cτaτ|ux, τ|2 dx
0
β
1
exp−cτaτ|ux, τ|2 dx
2.13
δ
τ1
≤ 16
px exp−ct|Lu|2 dxdt
0
α
0
1
a0|lu| dx 2
0
2
1
γ
1
dlu 2
2
pxa0
dx 2 a0|lu| dx a0|lu|2 dx.
dx 0
β
δ
Now, from 1.1, we have
c02
6
∂2 u 2
px exp−ct 2 dxdt
∂x 0
τ1
0
≤
τ1
0
px
exp−ct|Lu|2 dxdt 3
0
τ1
0
2
∂u px
exp−ct dxdt.
∂t
0 3
2.14
Combining inequalities 2.13 and 2.14, we get
exp−cT τ1
1
α
px ∂u 2
px ∂ux, τ 2
2
∂t dxdt c0
∂x dx c0 |ux, τ| dx
0 0 3
0 2
0
⎞
γ
1
2 τ1
∂2 u 2
c
px
2c0 |ux, τ|2 dx c0 |ux, τ|2 dx 0
dxdt⎠
6 0 0 3 ∂x2 β
δ
13
32
1
α
γ
1
px
px dlu 2
2
2
2
2
≤ 49
|Lu| dxdt c1
dx dx c1 |lu| dx 2c1 |lu| dx c1 |lu| dx.
0 2
0
β
δ
Ω 3
2.15
As the right-hand side of 2.15 is independent of τ, by replacing the left-hand side by its
upper bound with respect to τ in the interval 0, T , we obtain the desired inequality.
Journal of Applied Mathematics and Stochastic Analysis
9
3. Solvability of the Problem
From estimates 2.1 and 2.3, it follows that the operator L : E → F is continuous and its
range is closed in F. Therefore, the inverse operator L−1 exists and is continuous from the
closed subspace RL onto E, which means that L is an homeomorphism from E onto RL.
To obtain the uniqueness of solution, it remains to show that RL F. The proof is based on
the following lemma.
Lemma 3.1. Let
D0 L {u ∈ E : lu 0}.
3.1
If for u ∈ D0 L and some w ∈ L2 Ω, one has
Ω
qxLuwdxdt 0,
3.2
where
⎧
⎪
x,
⎪
⎪
⎨
qx γ − β,
⎪
⎪
⎪
⎩
1 − x,
x ∈ 0, α,
x ∈ α, δ,
3.3
x ∈ δ, 1,
then w 0.
Proof. From 3.2 we have
Ω
qx
∂u
wdxdt ∂t
Ω
qxat
∂2 u
wdxdt.
∂x2
3.4
Now, for given wx, t, we introduce the function
α
⎧
wξ, t
⎪
⎪
dξ, x ∈ 0, α,
wx,
t
−
⎪
⎪
ξ
⎪
x
⎪
⎪
⎨
x ∈ α, δ,
vx, t wx, t,
⎪
⎪
⎪
x
⎪
⎪
wξ, t
⎪
⎪
⎩wx, t −
dξ, x ∈ δ, 1.
ξ
δ
3.5
10
Journal of Applied Mathematics and Stochastic Analysis
Integrating by parts with respect to ξ, we obtain
α
⎧
⎪
⎪
vξ, tdξ,
xv
⎪
⎪
⎪
x
⎪
⎪
⎪
⎪
⎪
⎪
γ − β v,
⎪
⎨
γ
qxw ⎪
⎪
⎪
γ − β v vξ, tdξ,
⎪
⎪
⎪
β
⎪
⎪
x
⎪
⎪
⎪
⎪
⎩1 − xv vξ, tdξ,
x ∈ 0, α,
x ∈ α, β ∪ γ, δ ,
3.6
x ∈ β, γ ,
x ∈ δ, 1,
δ
which implies that
α
γ
1
vξ, tdξ 2 vξ, tdξ vξ, tdξ 0.
0
β
3.7
δ
Then, from 3.4, we obtain
−
∂u
Nvdxdt Ω ∂t
Ω
3.8
Atuvdxdt,
where
Nv qxv,
∂
∂u
Atu −
qxat
.
∂x
∂x
3.9
∗
−1
and I−1
If we introduce the smoothing operators with respect to t 16, I−1
ε I ε∂/∂t
ε ,
then these operators provide the solutions of the respective problems:
ε
dgε t
gε t gt,
dt
gε tt0 0,
3.10
dgε∗ t
gε∗ t gt,
dt
gε∗ ttT 0,
3.11
−ε
and also have the following properties: for any g ∈ L2 0, T , the functions gε I−1
ε g and
∗
1
∗
−1
g
are
in
W
0,
T
such
that
g
t
0
and
g
t
0.
Morever,
I
commutes
gε∗ I−1
ε
ε
ε
tT
2
T
T ε t0
with ∂/∂t, so 0 |gε − g|2 dt → 0 and 0 |gε∗ − g|2 dt → 0 for ε → 0.
t
Putting u 0 expcτvε∗ x, τdτ in 3.8, where the constant c satisfies cc0 − c3 −
εc32 /c0 ≥ 0, and using 3.11, we obtain
−
Ω
expctvε∗ Nvdxdt
∂u
Atu exp−ct dxdt − ε
∂t
Ω
Ω
Atu
∂vε∗
dxdt.
∂t
3.12
Journal of Applied Mathematics and Stochastic Analysis
11
Integrating by parts each term in the right-hand side of 3.12 and taking the real parts
yield
1
∂ux, T 2
dx
aT qx exp−cT ∂t 0
dat ∂u 2
qx exp−ct cat −
∂x dxdt;
dt
Ω
∂u ∂vε∗
dat
∂vε∗
dxdt Re ε
qx
dxdt
Re −ε Atu
∂t
∂x ∂x
Ω
Ω dt
∂u
2 Re Atu exp−ct dxdt ∂t
Ω
ε
Ω
at expctqx
∂vε∗
∂x
3.13
3.14
dxdt.
Using ε-inequalities, we obtain
2
∂u εc2
∂vε∗
Re −ε Atu
dxdt ≥ − 3 qx exp−ct dxdt.
∂t
2c0 Ω
∂x
Ω
3.15
Combining 3.13 and 3.15, we get
εc32 ∂u 2
∗
dxdt ≤ 0.
Re
expctvε Nvdxdt ≤ − qx exp−ct cc0 − c3 −
c0 ∂x Ω
Ω
3.16
From 3.16, we deduce that
Re
expctvε∗ Nvdxdt ≤ 0.
3.17
Ω
Then, for ε → 0, we obtain
Re
Ω
expctvNvdxdt Ω
qx expct|v|2 dxdt ≤ 0.
3.18
We conclude that v 0, hence w 0, which ends the proof of the the lemma.
Theorem 3.2. The range RL of L coincides with F.
Proof. Since F is a Hilbert space, we have RL F if and only if the relation
px
Lufdxdt Ω 3
1
px dlu dϕ
dx dx dx
0 2
α
0
luϕdx γ
luϕdx β
for arbitrary u ∈ E and f, ϕ ∈ F, implies that f 0 and ϕ 0.
1
δ
luϕdx 0,
3.19
12
Journal of Applied Mathematics and Stochastic Analysis
Putting u ∈ D0 L in 3.19, we conclude from Lemma 3.1 that θf 0, where
⎧
⎪
xf,
x ∈ 0, α,
⎪
⎪
⎨
θf γ − β f, x ∈ α, δ,
⎪
⎪
⎪
⎩
1 − xf, x ∈ δ, 1,
3.20
then f 0.
Taking u ∈ E in 3.19 yields
1
px dlu dϕ
dx dx dx
0 2
α
0
luϕdx γ
β
luϕdx 1
luϕdx 0.
3.21
δ
The range of the operator l is everywhere dense in Hilbert space with the norm
1
1/2
α
γ
1
2
2
2
px dϕ 2
ϕ dx ϕ dx ϕ dx
,
dx dx 0 2
0
β
δ
3.22
hence, ϕ 0.
Acknowledgment
The authors would like to thank the referee for helpful suggestions and comments.
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