Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 64, pp. 1–9.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
NONLOCAL SINGULAR PROBLEM WITH INTEGRAL
CONDITION FOR A SECOND-ORDER PARABOLIC EQUATION
AHMED LAKHDAR MARHOUNE, AMEUR MEMOU
Abstract. We prove the existence and uniqueness of a strong solution for a
parabolic singular equation in which we combine Dirichlet with integral boundary conditions given only on parts of the boundary. The proof uses a priori
estimate and the density of the range of the operator generated by the problem
considered.
1. Introduction
In the rectangle Ω = [0, 1] × [0, T ], we consider the equation
∂u 1 ∂
∂u £u =
−
x
= f (x, t),
∂t
x ∂x ∂x
with the initial condition
x ∈ [0, 1],
u(x, 0) = ϕ(x),
(1.1)
(1.2)
and the Dirichlet condition
t ∈ [0, T ],
u(1, t) = 0,
and the nonlocal condition
Z α
Z
u(x, t)dx +
0
(1.3)
1
u(x, t)dx = 0,
0 ≤ α ≤ β ≤ 1, t ∈ [0, T ].
(1.4)
β
The functions ϕ(x), f (x, t) are given, and we assume that the matching conditions
are satisfied
Z
0
α
ϕ(1) = 0,
Z 1
ϕ(x)dx +
ϕ(x)dx = 0.
β
Over the previous few years, many physical phenomena were formulated by means
of nonlocal mathematical models with integral boundary conditions. These integral
boundary conditions appear when the data on the body can not be measured directly, but their average values are known. For instance, in some cases, describing
the solution u (pressure, temperature, etc.) pointwise is not possible, because only
2000 Mathematics Subject Classification. 35B45, 35K67, 35K20.
Key words and phrases. Energy inequality; integral boundary conditions;
singular parabolic equation; strong solution.
c
2015
Texas State University - San Marcos.
Submitted September 3, 2014. Published March 18, 2015.
1
2
A. L. MARHOUNE, A. MEMOU
EJDE-2015/64
the average value of the solution can be estimate along the boundary or along a
part of it. These mathematical models are encountered in many engineering models
such as heat conduction [7], plasma physics [14], thermoelasticity [15], electrochemistry , chemical diffusion [3] and underground water flow [7, 17]. The importance
of this kind of problems have been also pointed out by Samarskii [14]. The first
paper, devoted to second order partial differential equations with nonlocal integral
conditions goes back to Cannon [4].This type of boundary value problems with
combined Dirichlet or Newmann and integral condition, or with purely integral
conditions has been investigated in [1, 2, 10, 19] for parabolic equations, for hyperbolic equations in [1, 13, 18], and in [6, 9] for mixed type equations. Problems for
elliptic equations with operator nonlocal conditions were considered by Mikhailov
and Gushin [8], A.L.Skubachevski, Steblov [16], Peneiah [12].
In this article we prove the existence and uniqueness of the strong solution of
a class of non local mixed second-order singular parabolic problem in which we
combine Dirichlet and integral conditions given only on parts of the boundary.
Case of α = 0, is treated in [5, 10]. This kind of problems for parabolic equations
was considered in [11].
2. Preliminaries
In this article, we prove the existence and uniqueness of a strong solution of
problem (1.1)-(1.4). For this, we consider the solution of problem (1.1)-(1.4) as a
solution of operator equation Lu = F = (f, ϕ), where the operator L is considred
from E to F , where E is the Banach space of the functions u, with the norm
Z 1 Z
∂2u 2
∂u 2
2 ∂u 2
x2 |u|2 + | |2 dx,
kukE =
x | | + | 2 | dx dt + sup
∂t
∂x
∂x
t
0
Ω
F is the Hilbert space of vector valued functions F = (f, ϕ) obtained by the completion of the space L2 (Ω) × W22 (0, 1), with respect to the norm
Z
Z 1 dϕ kF k2F =
x2 |f |2 dx dt +
x2 |ϕ|2 + | |2 dx,
dx
Ω
0
2
∂ u
with domain of definition D(L) consisting of functions u ∈ E, such that u, ∂u
∂x , ∂x∂t
belong to L2 (Ω) and u satisfies conditions (1.3)-(1.4). Then we establish an energy
inequality
kukE ≤ CkLukF , ∀u ∈ D(L),
(2.1)
and we show that the operator L has a closure L.
Definition 2.1. A solution of the operator equation Lu = F is called a strong
solution of problem (1.1)-(1.4).
Inequality (2.1), can be extended to u ∈ D(L), that is
kukE ≤ CkLukF ,
∀u ∈ D(L).
(2.2)
From this inequality, we obtain the uniqueness of a strong solution, if it exists, and
the equality of the sets R(L) and R(L). Thus, to prove the existence of the strong
solution of the problem (1.1)-(1.4) for any F ∈ F , it remains to prove that the set
R(L) is dense in F .
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NONLOCAL SINGULAR PROBLEM
3
3. An energy inequality and its applications
Theorem 3.1. There exists a positive constant C, such that, for any function
u ∈ D(L) we have
kukE ≤ CkLukF .
(3.1)
Proof. Let
Mu = x
2 ∂u
∂t
Z
−x
0
x
∂u
(ζ, t)dζ + x
∂t
Z
x
α
∂u
(ζ, t)dζ − x
∂t
Z
x
β
∂u
(ζ, t)dζ,
∂t
We consider the quadratic form obtained by multiplying (1.1) by exp(−ct)M u,
where c > 0 and integrating over Ωs = [0, 1] × [0, s] with 0 ≤ s ≤ T , and taking the
real part, formally
Φ(u, u)
Z
= Re
Ωs
exp(−ct)
∂u
M u dx dt −
∂t
Z
exp(−ct)
Ωs
∂u
1 ∂
(x )M u dx dt.
x ∂x ∂x
(3.2)
Integrating each term by parts in (3.2) with respect to x and using the condition
(1.4), we obtain
Z
∂u
Re
exp(−ct) M u dx dt
∂t
Ωs
Z
Z
Z
∂u
∂u x ∂u
=
x2 exp(−ct)| |2 dx dt +
exp(−ct)
ζ dζ
∂t
∂t 0 ∂t
Ωs
Ωs
(3.3)
Z
Z
Z 1
∂u
1 s
∂u
=
x2 exp(−ct)| |2 dx dt +
exp(−ct)dt|
x dx|2
∂t
2 0
∂t
Ωs
0
R x ∂u 2
Z
|ζ | dζ
+
exp(−ct) 0 ∂t2
dx dt .
2x
Ωs
Using conditions (1.3), (1.4), we obtain
Z
Z
∂u
1 ∂
∂u ∂ 2 u
(x )M u dx dt =
x2
dx dt.
−
exp(−ct)
x ∂x ∂x
∂x ∂x∂t
Ωs
Ωs
(3.4)
Integrating with respect to t, in the right hind side of (3.4), using (3.3), expression
(3.2) becomes
Z
Z
Z 1
Z R x ∂u 2
|ζ ∂t | dζ
1 s
∂u 2
2 ∂u 2
0
x | | dx dt +
dt|
x dx| +
dx dt
∂t
2 0
∂t
2x2
Ωs
0
Ωs
Z
Z 1 2
∂u
x
∂u
x2
(3.5)
exp(−ct)| |2 dx dt +
exp(−ct)| |2 dx dx
+c
2
∂x
2
∂x
t=s
0
Ωs
Z
Z 1 2
x dϕ 2
| | dx.
= Re
exp(−ct)£uM u dx dt +
Ωs
0 2 dx
Substituting M u by its expression in the first term in the right-hand side of (3.5),
we obtain
Z
Re
exp(−ct)£uM u dx dt
Ωs
Z
Z
Z x
Z x ∂u
∂u
∂u
= Re
x2 exp(−ct)f
dx dt − Re
xf
dζ −
dζ
∂t
∂t
Ωs
Ωs
0
α ∂t
4
A. L. MARHOUNE, A. MEMOU
Z
x
+
β
EJDE-2015/64
∂u dζ dx dt.
∂t
By integrating with respect to x, using the condition (1.4), we obtain
Z
Z x
Z x
Z x
Z
Z
∂u
∂u
∂u
∂u x
ζf dζ dx dt,
− Re
xf (
dζ −
dζ +
dζ) dx dt =
0 ∂t
α ∂t
β ∂t
Ωs
Ωs ∂t 0
then by using ε-inequalities, we have
Z
∂u
Re
x2 exp(−ct)f
dx dt
∂t
Ωs
Z
Z
ε1
1
∂u
≤
x2 exp(−ct)|f |2 dx dt +
x2 exp(−ct)| |2 dx dt,
2 Ωs
2ε1 Ωs
∂t
Z x
Z
∂u
ζf dζ dx dt
Ωs ∂t 0
Rx
Z
Z
| 0 ζf dζ|2
1
∂u 2
ε2
2
≤
x exp(−ct)| | dx dt +
exp(−ct)
dx dt
2ε2 Ωs
∂t
2 Ωs
x2
It is easy to show that
Rx
Z
Z
| 0 ζf (ζ, t)dζ|2
exp(−ct)
dx
dt
≤
4
x2 exp(−ct)|f |2 dx dt.
2
x
Ωs
Ωs
Then, from the previous inequalities, formula (3.5) becomes
Z
Z
Z 1
1
1
∂u
1 s
∂u
dt|
x dx|2
(1 −
−
)x2 exp(−ct)| |2 dx dt +
2ε1
2ε2
∂t
2 0
∂t
Ωs
0
Z R x ∂u 2
Z
2
|ζ ∂t | dζ
x
∂u
0
+
dx dt + c
exp(−ct)| |2 dx dt
2
2x
2
∂x
Ωs
Ωs
Z 1 2
x
∂u
+
exp(−ct)| |2 dx dx
∂x
t=s
0 2
Z
Z 1 2
ε1
x dϕ 2
≤ ( + 2ε2 )
x2 exp(−ct)|f |2 dx dt +
| | dx,
2
Ωs
0 2 dx
we choose ε1 = ε2 = 2, then
Z
Z
∂u
∂u
x2 exp(−ct)| |2 dx dt +
x2 exp(−ct)| |2 dx dt
∂t
∂x
Ωs
Ωs
Z 1
∂u
+
x2 exp(−ct)| |2 dx dx
∂x
t=s
0
Z
Z 1
10
dϕ
2
2
≤
x exp(−ct)|f | dx dt +
x2 | |2 dx .
min(1, c) Ωs
dx
0
Hence from (1.1), (3.6) we deduce that
Z
∂2u
x2 exp(−ct)| 2 |2 dx dt
∂x
Ωs
Z 1
Z
80
dϕ
x2 | |2 dx .
≤(
+ 2)
x2 exp(−ct)|f |2 dx dt +
min(1, c)
dx
0
Ωs
(3.6)
EJDE-2015/64
NONLOCAL SINGULAR PROBLEM
5
Integrating the term x2 exp(−ct)u ∂u
∂t with respect to t and using (3.6), we obtain
Z 1
x2 exp(−ct)|u|2 dx dx
t=s
0
10
≤(
+ 1)
c min(1, c)
Z
x2 exp(−ct)|f |2 dx dt +
Ωs
Z
1
x2 |ϕ|2 dx .
0
Then from the previous inequalities we obtain
Z 1 Z
∂u
∂2u 2
2 ∂u 2
x2 | |2 + |u|2 dx dx
x | | + | 2 | dx dt +
∂t
∂x
∂x
t=s
0
Ωs
Z 1
Z
dϕ
≤β
x2 |f |2 dx dt +
x2 (| |2 + |ϕ|2 )dx .
dx
Ω
0
(3.7)
The left side of (3.7) is independent of t, then by taking the upper bound with
respect to t from 0 to T , we obtain the desired inequality
kukE ≤ CkLukF , ∀u ∈ D(L),
where
C 2 = β = max
90
10
+3+
ecT .
min(1, c)
c min(1, c)
Lemma 3.2. The operator L from E to F admits a closure L.
The previous Theorem is valid for a strong solution, then we have the inequalities
kukE ≤ CkLukF , ∀u ∈ D(L).
Hence we obtain the following corollaries
Corollary 3.3. A strong solution of problem (1.1)-(1.4) is unique if it exists, and
depends continuously on F.
Corollary 3.4. The range R(L) of the operator L is closed in F , and R(L) = R(L).
4. Solvability of problem (1.1)-(1.4)
To prove the solvability of problem (1.1)-(1.4), it is sufficient to show that R(L)
is dense in F . The proof is based on the following lemma.
Lemma 4.1. Let D0 (L) = {u ∈ D(L), u(x, 0) = 0, }. If, for u ∈ D0 (L) and for
some function w ∈ L2 (Ω),
Z
φ(x)£uw dx dt = 0,
(4.1)
Ω
where
(
x3 ,
φ(x) =
x(x − β)2 ,
then w = 0.
x ∈ (0, α) ∪ (α, β),
x ∈ (β, 1),
6
A. L. MARHOUNE, A. MEMOU
EJDE-2015/64
Proof. Equality (4.1) can be written as
Z
Z
∂u
N v dx dt =
A(t)uv dx dt,
Ω ∂t
Ω
where

Rα

x ∈ (0, α),
xw − x wdζ,
v = x2 w,
x ∈ (α, β),

R1

(x − β)w − x wdζ, x ∈ (β, 1),
and
A(t)u =
(4.2)
(4.3)
∂
∂u xρ(x)
,
∂x
∂x
where


x ∈ (0, α),
x,
ρ(x) = 1,
x ∈ (α, β),


(x − β), x ∈ (β, 1),
and

Rx
2
3

x ∈ (0, α),
x v − x 0 vdζ = x w,
3
N v = xv = x w,
(4.4)
x ∈ (α, β),

x(x − β)v − x R x vdζ = x(x − β)2 w, x ∈ (β, 1).
β
Rα
R1
From (4.3), we conclude that 0 vdx + β vdx = 0.
We introduce the smoothing operators
∗
∂ −1
∂ −1
,
Jε−1 = I − ε
,
Jε−1 = I + ε
∂t
∂t
with respect to t, then, these operators provide the solution of the problems:
∂uε
= u(t) uε (0) = 0,
uε (t) − ε
∂t
∗
∂v
vε∗ (t) + ε ε = v(t) vε∗ (T ) = 0 .
∂t
We also have the following properties: for any g ∈ L2 (0, T ), the functions J−1 g,
(J−1 )∗ g ∈ W21 (0, T ). If g ∈ D(L), then J−1 g ∈ D(L) and we have
lim kJ−1 g − gkL2 (0,T ) = 0,
lim k(J−1 )∗ g
− gkL2 (0,T ) = 0,
for ε → 0,
for ε → 0 .
(4.5)
Substituting u in (4.2) by the smoothing function u and using the relation
A(t)u = J−1 Au,
we obtain
Z
∂vε∗
dx dt = −
A(t)uvε∗ dx dt.
(4.6)
∂t
Ω
Ω
The left-hand side of (4.6) is a continuous linear functional of u. Hence the function
∂v ∗
∂v ∗ ∂
(xρ(x) ∂xε ) ∈ L2 (Ω) and the following conditions
vε∗ has the derivatives xρ(x) ∂xε , ∂x
are satisfied:
vε∗ x=α = vε∗ x=β = vε∗ x=1 = 0,
(4.7)
∂vε∗ ∂v ∗ ∂v ∗ ∂v ∗ = ε
= x2 ε = ε
= 0.
∂x x=α
∂x x=β
∂x x=0
∂x x=1
Z
uN
EJDE-2015/64
NONLOCAL SINGULAR PROBLEM
Substituting u =
Rt
0
7
exp(−cτ )vε∗ dτ in (4.2), where the constant c < 0, we obtain
Z
Z
(4.8)
exp(−ct)vε∗ N v dx dt =
A(t)uv dx dt.
Ω
Ω
Using the properties of the smoothing operators we have
Z
Z
Z
∂v ∗
exp(−ct)vε∗ N v dx dt =
A(t)uvε∗ dx dt − A(t)u ε dx dt.
∂t
Ω
Ω
Ω
(4.9)
Integrating with respect to x and t, using (4.7) we obtain
Z
Re
A(t)uvε∗ dx dt
Ω
Z
=−
xρ(x) exp(ct)
Ω
1
Z
=−
0
∂u ∂ 2 u
dx dt
∂t ∂x∂t
xρ(x)
∂u
exp(ct)| |2 |t=T + c
2
∂x
Z
Ω
xρ(x)
∂u
exp(ct)| |2 dx dt ≤ 0.
2
∂x
Integrating by parts the second terms with respect to x and t in the right hand side
of (4.9) we obtain
Z
Z
∂u ∂ 2 vε∗
∂v ∗
xρ(x)
dx dt
−
A(t)u ε dx dt = ∂t
∂t ∂x∂t
Ω
Ω
(4.10)
Z
∂2u 2
= −
xρ(x) exp(ct)|
| dx dt ≤ 0.
∂x∂t
Ω
Substituting the expression of N v in (4.8), we obtain
Z
Z
Z
exp(−ct)vε∗ N v dx dt =
exp(−ct)(vε∗ − v)N v dx dt +
exp(−ct)vN v dx dt,
Ω
Ω
Ω
since
Z
exp(−ct)vN v dx dt
Ω
T
Z
=
0
Z
Z
α
exp(−ct)x2 |v|2 dx dt +
0
T Z 1
Z
0
T
Z
β
exp(−ct)x|v|2 dx dt
α
1
+
exp(−ct)x(x − β)|v| dx dt +
2
0
β
Z TZ 1 Z x
1
|
vdζ|2 dx dt,
+
2 0 β
β
2
Z
T
Z
α
Z
|
0
0
x
vdζ|2 dx dt
0
then by passing to the limit as ε −→ 0, we obtain v = 0, so that w = 0.
Theorem 4.2. The range R(L) of L coincides with F .
Proof. Since F is the Hilbert space, R(L) = F if and only if the relation
Z
Z 1
Z 1
dϕ dϕ1
x2 £uF 1 dx dt +
x2 ϕϕ1 dx +
x2
dx = 0,
dx dx
Ω
0
0
(4.11)
is satisfied for arbitrary u ∈ D(L) Rand F1 = (g, ϕ1 ) ∈ F imply F1 = 0. Taking
u ∈ D0 (L) in (4.11), we obtain that Ω x2 £uF 1 dx dt = 0 and using Lemma 4.1, we
8
A. L. MARHOUNE, A. MEMOU
EJDE-2015/64
obtain that φ(x)w = x2 g, then g = 0. Consequently for u ∈ D(L) we have
Z
0
1
x2 ϕϕ1 dx +
Z
0
1
x2
dϕ dϕ1
dx = 0,
dx dx
since the range of the operator trace is dense in the Hilbert space with the norm
Z 1 dϕ
x2 | |2 + |ϕ|2 dx.
dx
0
then ϕ = 0.
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EJDE-2015/64
NONLOCAL SINGULAR PROBLEM
9
Ahmed Lakhdar Marhoune
Laboratoire Equations Differentielles, Departement de Mathematiques, Faculte des
Sciences Exactes, Université des Fréres Mentouri, 25000 Constantine, Algerie
E-mail address: ahmed marhoune@hotmail.com
Ameur Memou
Departement de Mathematiques, Université de M’sila, Algerie
E-mail address: memoua 2007@yahoo.fr