Electronic Journal of Differential Equations, Vol. 2000(2000), No. 43, pp. 1–8.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
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Uniqueness of solutions to a system of differential
inclusions ∗
Chunpeng Wang & Jingxue Yin
Abstract
In this paper we study the uniqueness of solutions to the initial and
Dirichlet boundary-value problem of differential inclusions
→
∆ui + ∇· Bi (u1 , u2 , . . . , uN ) ∈
∂Fi (ui )
,
∂t
i = 1, 2, . . . , N,
→
where Bi (s1 , s2 , . . . , sN ) is an n-dimensional vector continuously differentiable on RN , and Fi (ui ) = {wi : ui = Ai (wi )}, i = 1, 2, . . . , N with
Ai (s) continuously differentiable functions on R and A0i (s) ≥ 0.
1
Introduction
This paper concerns with the system of differential inclusions
→
∆ui + ∇· Bi (u1 , u2 , . . . , uN ) ∈
∂Fi (ui )
,
∂t
(x, t) ∈ QT ,
i = 1, 2, . . . , N, (1.1)
where Ω is a bounded domain in Rn with smooth boundary ∂Ω, QT = Ω×(0, T ),
→
with T > 0, n and N are positive integers, Bi (s1 , s2 , . . . , sN ) is an n-dimensional
vector continuously differentiable on RN , and
Fi (ui ) = {wi : ui = Ai (wi )},
i = 1, 2, . . . , N
with Ai (s) continuously differentiable functions on R and A0i (s) ≥ 0. Note that
if Ai (s) is strictly increasing, then Fi (ui ) is single-valued, and (1.1) becomes
equality. However, we are interested in the case when some or all Ai (s)’s are
only nondecreasing, and so the Fi (ui )’s are interval-valued functions.
System (1.1) arises from mathematical models describing the nonlinear diffusion phenomena which exist in nature extensively. An important classical case
→
→
of (1.1) is that with Bi = 0 and N = 1. In this case (1.1) can be changed to
∂w
= ∆A(w).
∂t
∗ Mathematics Subject Classifications: 35K50, 35K65, 35A05, 35D99.
Key words: differential inclusions, degeneracy, uniqueness.
c
2000
Southwest Texas State University and University of North Texas.
Submitted May 15, 2000. Published June 12, 2000.
1
2
A system of differential inclusions
EJDE–2000/43
Brézis and Crandall [2] proved the uniqueness of bounded measurable solutions
for the Cauchy problem of the equation, where the nonlinear function A(s) is
assumed to be only non-decreasing. In other words, if A(s) is differentiable,
then
A0 (s) ≥ 0 ;
namely, the equation is permitted to be strongly degenerate. Thereafter some
authors tried to extend the uniqueness results to the equation with convection,
i.e.,
→
∂w
= ∆A(w) + ∇· B (w).
∂t
However, in most of those works, the nonlinear function A(s) is assumed to be
strictly increasing. In other words, the equation is weakly degenerate, see for
example [4, 3, 8, 9].
In this paper we study the uniqueness of solutions of the initial and Dirichlet
boundary-value problem of (1.1). The initial-boundary conditions are
ui = 0, (x, t) ∈ ∂Ω × [0, T ], i = 1, 2, . . . , N,
Fi (ui )(x, 0) = {fi (x)}, x ∈ Ω, i = 1, 2, . . . , N.
(1.2)
(1.3)
Definition For i = 1, 2, . . . , N , let fi ’s be bounded and measurable functions.
(u1 , u2 , . . . , uN ) is called a solution of the initial and Dirichlet boundary-value
problem (1.1)–(1.3), if the ui ’s are bounded and measurable functions and there
exist bounded measurable functions wi ∈ Fi (ui ) such that for arbitrary test
function ϕ in C ∞ (QT ) with value zero for x ∈ ∂Ω and for t = T , the following
integral equalities hold
ZZ
Q
ZT
+
Ω
→
∂ϕ
dx dt
ui ∆ϕ− Bi (u1 , u2 , . . . , uN ) · ∇ϕ + wi
∂t
fi (x)ϕ(x, 0)dx = 0,
i = 1, 2, . . . , N.
The main result of this paper is the following theorem.
Theorem 1 The initial and Dirichlet boundary-value problem (1.1)–(1.3) has
at most one solution.
The method of the proof is inspired by Brézis and Crandall [2]. Here what we
consider is not the Cauchy problem but the initial and Dirichlet boundary-value
problem, so we adopt the self-adjoint operators with homogeneous Dirichlet
boundary condition to prove the uniqueness instead of the self-adjoint operators
on the whole space. Moreover, the problem which we consider is a system of
differential inclusions with convection, so we must overcome some other technical
difficulties.
EJDE–2000/43
2
Chunpeng Wang & Jingxue Yin
3
Proof of the main theorem
We first introduce a family of operators. The L2 theory for elliptic equations
(see, e.g., [5] ) implies that for each λ > 0 and f ∈ H −1 (Ω), the Dirichlet
problem
−∆u + λu = f, x ∈ Ω,
u = 0, x ∈ ∂Ω,
(2.1)
(2.2)
has a unique solution u ∈ H01 (Ω). For 0 < λ < 1, we define the operator
Tλ : H −1 (Ω) → H01 (Ω),
f 7→ u,
where u is the unique solution to (2.1)–(2.2). It is easy to see that Tλ is selfadjoint, namely, for arbitrary f, g ∈ H −1 (Ω),
hf, Tλ gi = hg, Tλ f i
holds, where h·, ·i represents the dual product between H −1 (Ω) and H01 (Ω).
Specially, for f ∈ L2 (Ω) and g ∈ H −1 (Ω), we have
Z
f Tλ gdx.
hf, Tλ gi =
Ω
2
In addition, for arbitrary f ∈ L (Ω), the L2 theory for elliptic equations also
implies Tλ f ∈ H 2 (Ω) ∩ H01 (Ω) and
kTλ f kH 2 (Ω) ≤ C0 kf kL2 (Ω) ,
(2.3)
here C0 is a constant depending only on n and Ω, but independent of λ.
Proof of Theorem 1. Let (u1 , u2 , . . . , uN ) and (û1 , û2 , . . . , ûN ) be two solutions to (1.1)–(1.3). For i = 1, 2, . . . , N , the bounded measurable functions in
Fi (ui ) and Fi (ûi ) satisfying the definition are denoted by wi and ŵi correspondingly. For i = 1, 2, . . . , N , we set
→
→
Hi =Bi
vi = ui − ûi ,
zi = wi − ŵi ,
→
(u1 , u2 , . . . , uN )− Bi (û1 , û2 , . . . , ûN ).
→
The definition of solutions implies that zi , vi and Hi (i = 1, 2, . . . , N ) are
all bounded measurable functions, and for arbitrary test function ϕ, namely,
ϕ ∈ C ∞ (QT ) with ϕ = 0 for x ∈ ∂Ω and for t = T , the integral equalities
ZZ →
∂ϕ
dx dt = 0, i = 1, 2, . . . , N
(2.4)
vi ∆ϕ− Hi ·∇ϕ + zi
∂t
QT
hold. Let ψ ∈ C ∞ ([0, T ]) with ψ(T ) = 0 and k ∈ C0∞ (Ω). Then we see that
Tλ k ∈ H 2 (Ω) ∩ H01 (Ω). By an approximate process, we may choose ψTλ k as a
test function. Letting ϕ = ψTλ k in (2.4), we get
ZZ →
∂ψ
zi Tλ k dx dt = 0.
λψvi Tλ k − ψvi k − ψ Hi ·∇Tλ k +
∂t
QT
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A system of differential inclusions
EJDE–2000/43
Using integration by parts and the self-adjointness of Tλ , we get
ZZ →
∂Tλ zi
dx dt = 0.
λψkTλ vi − ψkvi + ψkTλ (∇· Hi ) − ψk
∂t
QT
Owing to the arbitrariness of ψ and k, we see that
→
∂Tλ zi
= λTλ vi − vi + Tλ (∇· Hi )
∂t
(2.5)
λ zi
∈ L2 (QT ) and Tλ zi ∈ H 2 (Ω) ∩
in the sense of distribution. It follows that ∂T∂t
1
∞
H0 (Ω). Let ψ ∈ C ([0, T ]) with ψ(T ) = 0. By an approximate process, we
may choose ψTλ zi as a test function. Letting ϕ = ψTλ zi in (2.4), we get
ZZ →
∂ψ
∂Tλ zi
zi Tλ zi + ψzi
dx dt = 0.
λψvi Tλ zi − ψvi zi − ψ Hi ·∇Tλ zi +
∂t
∂t
QT
(2.6)
Combining (2.5) with (2.6), we see that
ZZ →
λψvi Tλ zi + λψzi Tλ vi − 2ψvi zi + ψzi Tλ (∇· Hi )
QT
→
−ψ Hi ·∇Tλ zi +
∂ψ
zi Tλ zi dx dt =
∂t
0.
Using integration by parts and the self-adjointness of Tλ , for i = 1, 2, . . . , N , we
get
ZZ →
∂ψ
zi Tλ zi dx dt = 0.
2λψvi Tλ zi − 2ψvi zi − 2ψ Hi ·∇Tλ zi +
(2.7)
∂t
QT
Z
Let
giλ (t) =
Ω
zi Tλ zi dx,
t ∈ [0, T ],
i = 1, 2, . . . , N.
Now we prove that giλ (t) converges to zero on [0, T ] uniformly as λ → 0 for
i = 1, 2, . . . , N .
First, we show that giλ (t) is absolutely continuous. From (2.7), we get
ZZ →
∂(zi Tλ zi )
dx dt = 0.
2λψvi Tλ zi − 2ψvi zi − 2ψ Hi ·∇Tλ zi − ψ
∂t
QT
From the arbitrariness of ψ, we see that
Z
Z
d
∂(zi Tλ zi )
0
dx
giλ
(t) =
zi Tλ zi dx =
dt Ω
∂t
Ω
Z
Z
Z →
= 2λ
vi Tλ zi dx − 2
vi zi dx − 2
Hi ·∇Tλ zi dx,
Ω
→
Ω
Ω
a.e. t ∈ [0, T ].
Since zi , vi and Hi are all bounded measurable functions and (2.3) holds, we
0
get that giλ
(t) ∈ L1 (0, T ). Thus giλ (t) is absolutely continuous.
EJDE–2000/43
Chunpeng Wang & Jingxue Yin
5
Next, we show that giλ (0 + 0) ≡ limt→0+ giλ (t) = 0. Let
Z +∞
1 s
,
αε (s − ε)ds, αε (s) = α
ψε (t) =
ε
ε
t
where α(s) denotes the kernel of one-dimensional mollifier, namely, α is in the
R1
space C0∞ (−∞, +∞), α ≥ 0, suppα = [−1, 1] and −1 α(s) ds = 1. Thus ψε ∈
C ∞ ([0, T ]) and ψε (T ) = 0 for sufficiently small ε > 0. Letting ψ = ψε in (2.7),
we get
ZZ
→
(2λψε vi Tλ zi − 2ψε vi zi − 2ψε Hi ·∇Tλ zi − αε (t − ε)zi Tλ zi ) dx dt = 0.
QT
The dominated convergence theorem implies
Z 2ε
αε (t − ε)giλ (t)dt
giλ (0 + 0) = lim+
ε→0
0
ZZ
αε (t − ε)zi Tλ zi dx dt
= lim
ε→0+
QT
ZZ
ZZ
ψε vi Tλ zi dx dt − 2 lim+
ψε vi zi dx dt
= 2λ lim+
ε→0
ε→0
QT
QT
ZZ
→
−2 lim+
ψε Hi ·∇Tλ zi dx dt .
ε→0
QT
→
Hi
are all bounded measurable functions and (2.3) holds, we
Since zi , vi and
get that giλ (0 + 0) = 0.
Finally, we prove that giλ (t) converges to zero on [0, T ] uniformly as λ → 0.
It follows easily from the above arguments that
Z t
0
giλ (t) = giλ (0 + 0) +
giλ
(s)ds
0
Z tZ
Z tZ
vi Tλ zi dx ds − 2
vi zi dx ds
= 2λ
0
−2
Ω
Z tZ
0
Ω
0
→
Hi
Ω
·∇Tλ zi dx ds .
→
Since wi and ŵi are bounded measurable and Ai and Bi are continuously differentiable, there exist three positive constants M0 , M1 and M2 such that for
i = 1, 2, . . . , N , the following estimates hold

1/2
N
X
→
vj2  .
|zi | ≤ M0 , |vi | ≤ M1 |zi |, | Hi | ≤ M2 
j=1
Noticing that zi and vi have the same sign for A0i (s) ≥ 0, we get
vi zi = |vi ||zi | ≥
1 2
v .
M1 i
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A system of differential inclusions
EJDE–2000/43
By Schwarz’s inequality and Young’s inequality, we get
Z →
Hi ·∇Tλ zi dx
Ω
Z
≤
1/2 Z
Ω
≤ M2
|
→
Hi |2 dx
N Z
X
j=1
≤
=
=
≤
1
N M1
Ω
Ω
1/2 Z
1/2
2
2
vj dx
|∇Tλ zi | dx
N Z
X
j=1
1/2
|∇Tλ zi | dx
2
Ω
Ω
vj2 dx
N 2 M1 M22
+
4
Z
Ω
(∇Tλ zi ∇Tλ zi )dx
Z
N Z
N 2 M1 M22
1 X
2
v dx +
(−Tλ zi ∆Tλ zi )dx
N M1 j=1 Ω j
4
Ω
Z
N Z
N 2 M1 M22
1 X
2
vj dx +
−λ(Tλ zi )2 + zi Tλ zi dx
N M1 j=1 Ω
4
Ω
N Z
N 2 M1 M22
1 X
giλ (t).
vj2 dx +
N M1 j=1 Ω
4
Let
gλ (t) =
N
X
giλ (t).
i=1
Therefore,
gλ (t)
=
2
Z tZ
Z tZ →
N Z tZ
X
λ
vi Tλ zi dx ds −
vi zi dx ds −
Hi ·∇Tλ zi dx ds
≤
≤
Ω
0
i=1
0
Ω
0
Z tZ
N Z tZ
X
1
λ
vi Tλ zi dx ds −
vi2 dx ds
M
1
Ω
Ω
0
0
i=1
Z
N Z Z
N 2 M1 M22 t
1 X t
2
vj dx ds +
giλ (s)ds
+
N M1 j=1 0 Ω
4
0
2
2λ
N Z
X
i=1
0
T
Z
N 2 M1 M22
|vi Tλ zi | dx ds +
2
Ω
Moreover, it follows that
gλ (t) ≥ 0
by
Z
giλ (t) =
Ω
zi Tλ zi dx
Z
t
0
gλ (s)ds.
Ω
EJDE–2000/43
Chunpeng Wang & Jingxue Yin
7
Z
=
ZΩ
=
Ω
(−∆Tλ zi Tλ zi + λTλ zi Tλ zi )dx
(∇Tλ zi ∇Tλ zi + λTλ zi Tλ zi )dx
≥ 0.
Hence by Gronwall’s inequality, we get
gλ (t) ≤ C1 λ,
where C1 is a constant depending only on N , M0 , M1 , M2 , C0 , T and the
measure of Ω, but independent of λ and t. So gλ (t) converges to zero on [0, T ]
uniformly as λ → 0. Noticing that giλ (t) ≥ 0, we get that giλ (t) converges to
zero on [0, T ] uniformly as λ → 0.
Now we prove
zi (x, t) = 0,
a.e. (x, t) ∈ QT ,
i = 1, 2, . . . , N.
For any ϕ ∈ C0∞ (QT ), we have
ZZ
2
zi ϕ dx dt
QT
=
=
ZZ
2
(−∆Tλ zi + λTλ zi )ϕ dx dt
QT
ZZ
2
(∇T
z
∇ϕ
+
λϕT
z
)
dx
dt
λ i
λ i
QT
=
2k∇ϕk2L2 (QT ) k∇Tλ zi k2L2 (QT ) + 2λ2 kϕk2L2 (QT ) kTλ zi k2L2 (QT )
ZZ
ZZ
2
2
C2
|∇Tλ zi | dx dt + λ
(Tλ zi ) dx dt
QT
QT
ZZ
(−∆Tλ zi + λTλ zi )Tλ zi dx dt
C2
Q
ZZ T
C2
zi Tλ zi dx dt
≤
C2 T sup giλ (t) → 0,
≤
≤
≤
QT
t∈[0,T ]
(λ → 0),
where C2 = 2k∇ϕk2L2 (QT ) + 2kϕk2L2(QT ) independent of λ. Therefore,
ZZ
QT
zi ϕ dx dt = 0,
∀ϕ ∈ C0∞ (QT ).
It follows that
zi (x, t) = 0,
a.e. (x, t) ∈ QT ,
i = 1, 2, . . . , N.
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A system of differential inclusions
EJDE–2000/43
Thus
wi (x, t) = ŵi (x, t),
a.e. (x, t) ∈ QT ,
i = 1, 2, . . . , N,
a.e. (x, t) ∈ QT ,
i = 1, 2, . . . , N.
which implies
ui (x, t) = ûi (x, t),
The proof is complete.
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Chunpeng Wang & Jingxue Yin
Department of Mathematics, JiLin University,
Changchun, Jilin 130023, People’s Republic of China
e-mail: yjx@mail.jlu.edu.cn