Hindawi Publishing Corporation
Boundary Value Problems
Volume 2009, Article ID 739097, 23 pages
doi:10.1155/2009/739097
Research Article
On Initial Boundary Value Problems
with Equivalued Surface for Nonlinear
Parabolic Equations
Fengquan Li
Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China
Correspondence should be addressed to Fengquan Li, [email protected]
Received 6 January 2009; Revised 12 March 2009; Accepted 22 May 2009
Recommended by Sandro Salsa
We will use the concept of renormalized solution to initial boundary value problems with
equivalued surface for nonlinear parabolic equations, discuss the existence and uniqueness of
renormalized solution, and give the relation between renormalized solutions and weak solutions.
Copyright q 2009 Fengquan Li. 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
Let Ω ⊂ R
N
N ≥ 2 be a bounded domain with Lipschitz boundary ∂ΩΓ. T is a fixed
positive constant, Q Ω× 0,T. We consider the following nonlinear parabolic boundary
value problems with equivalued surface:
∂u
∂t
−
N
i,j1
Γ
∂u
∂n
L
ds A
t
∀ a.e.t∈
0,T
,
u
x, 0
0inΩ,
P
where f ∈ L
2
Q and A ∈ L
2
0,T, n n
1
, ,n
N
denotes the unit outward normal vector
on Γ and
∂u
the hypotheses of f ∈ L
q
Q and A ∈ L
r
0,T with q>N/2 1, r>N 2. However, if
f ∈ L
2
Q and A ∈ L
2
0,T, we cannot get a bounded weak solution. In order to deal with
this situation, we will introduce the concept of renormalized solution to problem P and
discuss the existence and uniqueness of renormalized solution.
The paper is organized as follows. In Section 2, we introduce the concept of
renormalized solution and prove the existence of renormalized solution to problem P.In
Section 3, uniqueness and a comparison principle of renormalized solution to problem P are
established. In Section 4, we discuss the relation between renormalized solutions and weak
solutions for problem P.
2. Existence of Renormalized Solution to Problem P
In order to prove the existence of renormalized solution to problem P, we make the
following assumptions.
Let a
ij
: Ω × R → R be Carath
´
eodory functions with 1 ≤ i, j ≤ N. We assume that
a
ij
·, 0 ∈ L
∞
Ω and for any given M>0 there exist d
≤ d
M
x
|
s
1
− s
2
|
,
|
s
k
|
≤ M, k 1, 2, 2.1
N
i,j1
a
ij
x, s
ξ
i
ξ
weak solutions to problem P, hence a
ij
·,uD
j
u may not belong to L
2
Q.Inorderto
overcome this difficulty, we will use the concept of renormalized solution introduced by
DiPerna and Lions in 9 for Boltzmann equations see also 10–12.
Boundary Value Problems 3
As usual, for k>0, T
k
denotes the truncation function defined by
T
k
v
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
k, if v>k,
v, if
an arbitrary function of t
. 2.5
Definition 2.1. A renormalized solution to problem P is a measurable function u : Q → R,
satisfying u ∈ L
2
0,T; V ∩ L
∞
0,T; L
2
Ω and for all h ∈ C
1
c
R, ξ ∈ W,
−
Q
ξ
t
u
0
h
r
drdxdt
Q
0
A
t
h
u
t
|
Γ
ξ
t
|
Γ
dt,
2.6
lim
m → ∞
{
x,t
∈Q:m≤
2
Q. The second term on the left
side of 2.6 should be understood as
{
x,t
∈Q:
|
u
|
<k
}
N
i,j1
a
ij
x, T
k
u
D
j
T
k
u
x,t
|
≤m1
}
N
i,j1
a
ij
x, T
m1
u
D
j
T
m1
u
D
i
T
m1
1,∞
R with compact support and all
ξ ∈{ξ ∈ L
2
0,T; V | ξ
t
∈ L
2
Q,ξ·,T0}.
4 Boundary Value Problems
Now we can state the existence result for prolem P as follows.
Theorem 2.5. Under hypotheses 2.1-2.2 and f ∈ L
2
Q, A ∈ L
2
0,T, problem P admits a
renormalized solution u ∈ L
2
0,T; V ∩ L
∞
0,T; L
2
Ω in the sense of Definition 2.1.
In order to prove Theorem 2.5, we will consider the following problem:
∂u
n
∂t
−
N
on Γ ×
0,T
,
Γ
∂u
n
∂n
L
ds A
t
∀ a.e.t∈
0,T
,
u
n
x, 0
0inΩ,
P
n
t
,v
V
,V
Ω
N
i,j1
a
n
ij
x, u
n
D
j
u
n
D
i
vdx
via Galerkin
method. Let us consider the operator
B : L
2
Ω
−→ V,
F −→ v,
2.12
where v is the weak solution of the following problem:
−Δv v F in Ω,
v C
a constant to be determined
on Γ,
Γ
∂v
∂n
L
ds 0.
E
By Lax-Milgram Theorem, the above problem exists a unique weak solution v which
continuously depending on F. Hence B is a compact self-adjoint operator from L
2
Ω to
Boundary Value Problems 5
L
wD
i
vdx,
Ft,v
V
,V
Ω
f
x, t
v
x
dx A
t
v|
Γ
, a.e.t∈
0,T
Au
m
n
t
,w
k
F
t
,w
k
, a.e.t∈
0,T
,
u
m
n
x, 0
0a.e.x∈ Ω.
≤ C
0
,
u
m
n
L
2
0,T;V
≤ C
0
,
Au
m
n
L
2
0,T;V
≤ C
0
,
m
n
u
n
weak ∗ in L
∞
0,T; L
2
Ω
,
u
m
n
u
n
weakly in L
2
0,T; V
,
u
m
n
, we omit the details.
6 Boundary Value Problems
To deal with the time derivative of truncation function, we introduce a time
regularization of a function u ∈ L
2
0,T; V .Let
u
ν
x, t
t
−∞
νu
x, s
e
ν
s−t
ds, u
x, s
u
u − u
ν
, 2.18
and finally if u ∈ L
∞
Q, then u
ν
∈ L
∞
Q and
u
ν
L
∞
Q
≤
u
L
∞
Q
, ∀ν>0. 2.19
Taking v u
n
t in 2.10, then integrating over 0,τ with τ ∈ 0,T, we have
n
D
j
u
n
D
i
u
n
dxdt
τ
0
Ω
fu
n
dxdt
τ
0
A
t
u
n
2
0,T;V
≤ C
1
, 2.22
where C
1
is a positive constant depending only on f
L
2
Q
, A
L
2
0,T
, λ
0
, but independent of
n and u
n
.
By 2.21 and 2.22, there is a subsequence of {u
n
} still denoted by {u
n
} such that
u
n
Using the same method as 10, we can obtain
u
n
−→ u a.e. in Q
up to some subsequence
. 2.24
Boundary Value Problems 7
Thus for any given k>0,
T
k
u
n
T
k
u
weakly in L
2
0,T; V
, strongly in L
2
Q
u
n
|
Γ
−→ u|
Γ
, a.e. in
0,T
. 2.28
For any given k>0, it follows from 2.27-2.28 and Vitali’s t heorem that
T
k
u
n
|
Γ
−→ T
k
u|
Γ
strongly in L
2
0,T
u|≤k,
η
ν
u
−→ T
k
u
strongly in L
2
0,T; V
, as ν tends to the infinity.
2.31
For any fixed h and k with h>k>0, let
w
n
T
2k
u
n
− T
h
u
n, ν, h
, 2.33
where lim
h → ∞
lim
ν →∞
lim
n → ∞
ωn, ν, h0.
Proof. The proof of Lemma 2.6 is the same as 10, Lemma 2.1, and we omit the details.
8 Boundary Value Problems
Lemma 2.7. Under the previous assumptions, for any given k>0, we have
T
k
u
n
−→ T
k
u
strongly in L
2
0,T; V
. 2.34
dxdt
T
0
A
t
w
n
t
|
Γ
dt ω
n, ν, h
. 2.35
Now note that Dw
n
0if|u
n
| >h 4k; then if we set M h 4k, splitting the integral
ontheleftsideof2.35 on the sets {x, t ∈ Q : |u
n
x, t| >k} and {x, t ∈ Q : |u
n
x, t|≤k},
a
ij
x, T
M
u
n
D
j
T
M
u
n
D
i
w
n
dxdt
≥
Q
N
i,j1
a
ij
dxdt
−
{|
u
n
|
>k
}
N
i,j1
a
ij
x, T
M
u
n
D
j
T
M
u
n
ij
x, T
M
u
n
D
j
T
M
u
n
D
i
η
ν
u
dxdt
|
D
i
T
k
u
|
dxdt
Q
N
i,j1
a
ij
x, T
M
u
n
D
ij
x, T
M
u
n
D
j
T
M
u
n
is bounded in L
2
Q
with respect to n, while |D
i
T
k
u|χ
{|u
n
|>k}
strongly converges to zero in L
2
Q. Moreover it
follows from 2.31 that
{|
u
n
i
η
ν
u
dxdt ≤ ω
n, ν
, 2.38
Boundary Value Problems 9
where lim
ν →∞
lim
n → ∞
ωn, ν0. Equations 2.38, 2.36,and2.35 imply that
Q
N
i,j1
a
ij
x, T
k
fw
n
dxdt
T
0
A
t
w
n
t
|
Γ
dt ω
n, ν
ω
n, ν, h
.
2.39
By 2.25, 2.31,and2.39,weget
Q
k
u
dxdt
≤
Q
fw
n
dxdt
T
0
A
t
w
n
t
|
Γ
dt ω
n, ν
ω
dxdt ω
n
, 2.41
where lim
n → ∞
ωn0. 2.31 and 2.41 imply that
Q
fw
n
dxdt
Q
fT
2k
u − T
h
u
dxdt ω
n, ν
; 2.42
Therefore we get
Q
N
i,j1
a
ij
x, T
k
u
n
D
j
T
k
u
n
− D
j
T
k
u
x, T
k
u
n
D
j
T
k
u
D
i
T
k
u
n
− T
k
u
dxdt.
2.45
10 Boundary Value Problems
D
i
T
k
u
n
− T
k
u
dxdt 0. 2.46
Using 2.2, 2.25,and2.46,weobtain2.34.
Proof of Theorem 2.5. For any given ξ ∈ W, h ∈ C
1
c
R, suppose that supp h ⊂ −k, k, taking
v hu
n
tξt in 2.10 and integrating over 0,T, w e have
T
0
u
h
u
n
ξ
dxdt
T
0
A
t
h
u
n
t
|
Γ
ξ
t
Q
ξ
t
u
n
0
h
r
drdxdt. 2.48
However
−
Q
ξ
t
u
n
0
h
r
drdxdt −→ −
Q
Q
ξ
t
u
0
h
r
drdxdt
Q
ξ
t
u
n
L
2
Q
−→ 0, as n −→ ∞.
2.50
Boundary Value Problems 11
As n>k, we have
Q
N
i,j1
a
n
ij
x, u
n
D
j
u
n
D
i
h
u
n
k
u
n
h
u
n
ξdxdt
Q
N
i,j1
a
ij
x, T
k
u
n
D
j
T
j
T
k
u
n
D
i
T
k
u
n
h
u
n
ξdxdt
−→
Q
N
i,j1
a
ij
i,j1
a
ij
x, T
k
u
n
D
j
T
k
u
n
D
i
ξh
u
n
dxdt
−→
Q
N
i,j1
a
n
ij
x, u
n
D
j
u
n
D
i
h
u
n
ξ
dxdt
−→
Q
N
i,j1
a
a
ij
x, u
D
j
uD
i
h
u
ξ
dxdt.
2.53
It follows from 2.28 that
T
0
A
t
h
u
n
t
|
Γ
dt. 2.54
12 Boundary Value Problems
Equation 2.24 yields
Q
fh
u
n
ξdxdt −→
Q
fh
u
ξdxdt. 2.55
Taking n → ∞ in 2.47,by2.48, 2.49,and2.53–2.55,weobtain
−
Q
ξ
t
T
0
A
t
h
u
t
|
Γ
ξ
t
|
Γ
dt
Q
fh
u
ξdxdt.
>dt
Q
N
i,j1
a
n
ij
x, u
n
D
j
u
n
D
i
T
1
u
n
− T
m
u
n
dt
Q
fT
1
u
n
− T
m
u
n
dxdt.
2.57
Setting S
1
s
s
0
T
1
t − T
m
tdt, then 0 ≤ S
1
s ≤|s|sign
i
u
n
dxdt ≤
{t∈0,T:|u
n
t|
Γ
|≥m}
|
A
t
|
dt
{
|u
n
|≥m
}
f
dxdt.
2.58
Boundary Value Problems 13
Only simply modifying 12, Lemma 3.1, we can obtain the following result.
Lemma 3.1. Let u be a renormalized solution to problem P for the data f, A.Then
Q
ξ
t
sign
0
u
u
0
h
r
drdxdt
T
0
sign
0
u
u
fh
u
ξdxdt
≥
Q
sign
0
u
N
i,j1
a
ij
x, u
D
j
uD
i
0
−u
t
|
Γ
A
t
ξ
t
|
Γ
h
u
t
|
Γ
dt
D
j
uD
i
h
u
ξ
dxdt,
3.2
for any h ∈ C
1
c
R,h≥ 0,ξ ∈ W, ξ ≥ 0.
Let sign
denote the multivalued function defined by sign
r0ifr<0, and
sign
0 ⊂ 0, 1,sign
r1ifr>0.
Lemma 3.2. For i 1, 2,letf
i
∈ L
|
Γ
such that
−
Q
ξ
t
sign
0
u
1
− u
2
u
1
u
2
h
r
drdxdt
Q
u
2
a
ij
x, u
2
D
j
u
2
D
i
ξdxdt
Q
N
i,j1
sign
0
u
1
− u
ij
x, u
2
D
j
u
2
D
i
u
2
ξdxdt
≤
Q
K
1
h
u
1
f
1
− h
u
2
t
|
Γ
A
2
t
ξ|
Γ
dt,
∀h ∈ C
1
c
R
,h≥ 0, ∀ξ ∈ W, ξ ≥ 0.
3.3
Proof. Let ξ ∈ W, ξ ≥ 0, ρ
l
be a sequence of mollifiers in R with supp ρ
l
⊂ −2/l, 0 and ρ
l
,
x, t
−→ ξ
l
x, t, s
∈ W, ∀s ∈
0,T
.
3.5
Let h ∈ C
1
c
R, h ≥ 0, H
ε
∈ W
1,∞
R be defined by H
ε
rHr/ε, where H ∈ W
1,∞
R,
Hr0forr ≤ 0, Hrr for 0 <r<1andHr1ifr ≥ 1. As u
1
f
1
h
u
1
H
ε
u
1
s, x
− u
2
t, x
ξ
l
dxds
T
0
A
1
ξ
l
s
|
Γ
ds
Q
N
i,j1
a
ij
x, u
1
D
j
u
1
D
i
h
u
2
0
h
r
H
ε
u
1
s, x
− r
drdxdt
Q
f
2
h
u
2
H
ε
u
H
ε
u
1
s
|
Γ
− u
2
t
|
Γ
ξ
l
t
|
Γ
dt
Q
2
t, x
ξ
l
dxdt.
3.7
Integrating the above two equalities in t, respectively, s over 0,T and taking their difference,
we get
T
0
Q
ξ
l
s
u
1
0
h
r
− r
dr
dxdsdt
T
0
Q
f
1
h
u
1
H
ε
u
1
s, x
− u
2
T
0
A
1
s
h
u
1
s
|
Γ
H
ε
u
1
s
|
Γ
− u
ε
u
1
s
|
Γ
− u
2
t
|
Γ
ξ
l
t
|
Γ
dsdt
Boundary Value Problems 15
T
s, x
− u
2
t, x
ξ
l
−a
ij
x, u
2
D
j
u
2
h
u
2
D
i
H
u
1
a
ij
x, u
1
D
j
u
1
D
i
u
1
H
ε
u
1
s, x
− u
2
t, x
t, x
ξ
l
dxdsdt.
3.8
Denote the three integrals on the left-hand side by I
1
, I
2
, I
3
, the two integrals on the right-
hand side by I
4
, I
5
.
It is easy to prove that
lim
l →∞
lim
ε → 0
I
5
Q
D
i
u
1
− h
u
2
a
ij
x, u
2
D
j
u
2
D
i
u
2
dxdt.
3.9
Similarly to the estimates for I
2
in 12, c.f. page 102, we can obtain
lim
ε → 0
I
3
≤
T
0
ξ
t
|
Γ
K
2
h
u
1
t
|
Γ
A
1
u
2
}
sign
0
f
1
− f
2
, K
2
χ
{t∈0,T:u
1
t|
Γ
>u
2
t|
Γ
}
sign
0
A
1
−
A
2
u
1
x, s
− u
2
x, t
ξ
l
s
u
1
x,s
u
2
x,t
h
r
dr
ξ
2
x, t
0
u
2
x,t
h
r
drdxdt
Q
ξ
l
x, 0,s
sign
0
u
1
x, s
ξ
t
x, t
ρ
l
t − s
u
1
x,s
u
2
x,t
h
r
drdsdt
Q
ξ
x, 0
We have
lim
l →∞
I
11
Q
sign
0
u
1
− u
2
ξ
t
u
1
u
2
h
r
drdxdt. 3.12
Consider the function
. 3.13
Note that ξ ∈ W,thusforl sufficiently large, φ
l
∈ W. Applying 3.1 with u u
1
, ξ φ
l
,
f f
1
, AtA
1
s,andt s, we have
I
12
−
Q
φ
l
s
sign
0
u
1
φ
l
s
|
Γ
h
u
1
s
|
Γ
ds
Q
sign
0
u
1
φ
i
φ
l
h
u
1
dxds.
3.14
It is easy to prove that the integrals on the right-hand side of 3.14 converge to 0 as l → ∞.
Thus we get
lim
l →∞
I
12
≤ 0. 3.15
It remains to consider I
4
. We have
I
4
T
0
Q
u
2
a
ij
x, u
2
D
j
u
2
H
ε
u
1
s, ·
− u
2
D
i
ξ
l
i
u
1
− u
2
t, ·
−h
u
2
a
ij
x, u
2
D
j
u
2
D
i
u
1
u
1
− u
2
N
i,j1
h
u
1
a
ij
x, u
1
D
j
u
1
− h
u
2
a
i,j1
h
u
1
x, s
− h
u
2
x, t
× a
ij
x, u
1
D
j
u
1
D
i
u
a
ij
x, u
1
D
j
u
1
− a
ij
x, u
2
D
j
u
2
D
i
u
1
− u
2
dxdsdt
l
|
h
|
d
k1
x
|
k 1 a
ij
x, 0
×
2
|
DT
k1
1
ε
0,T×Q∩{0<
|
u
1
x,s−u
2
x,t
|
<ε}
ξ
l
×
N
i,j1
h
u
2
x, t
a
ij
0,T×Q∩{0<
|
u
1
x,s−u
2
x,t
|
<ε}
ξ
l
18 Boundary Value Problems
×
N
i,j1
h
u
2
x, t
a
ij
x, u
1
2
x,t
|
<ε}
ξ
l
×
N
i,j1
h
u
2
|
d
k1
x
|
|
DT
k1
u
1
|
2
x,t
|
<ε}
ξ
l
×
N
i,j1
h
u
2
|
d
k1
x
|
|
DT
k1
u
1
|
W
1,∞
Q | ξT0,ξt|
Γ
Ctan arbitary function of t} and ξ ≥ 0.
Now we state the uniqueness and comparison principle of renormalized solution to
problem P as follows.
Theorem 3.4. Under hypotheses 2.1 and 2.2 ,fori 1, 2,letf
i
∈ L
2
Q, A
i
∈ L
2
0,T, u
i
be a
renormalized solution to problem P for the data f
i
,A
i
. Then there exist K
1
∈ sign
u
1
− u
2
K
1
f
1
− f
2
dxdt
τ
0
K
2
A
1
− A
2
dt. 3.24
In particular, for any given A ∈ L
2
0,T and f ∈ L
2
Q, the renormalized solution u to
problem P is unique.
Boundary Value Problems 19
Proof. For any given τ ∈ 0,T and any given ε>0sufficiently small, let α
ε
ε
in 3.3,weget
−
Q
α
ε
t
sign
0
u
1
− u
2
u
1
u
2
h
r
drdxdt
j
u
1
D
i
u
1
− h
u
2
a
ij
x, u
2
D
j
u
2
D
i
u
2
dxdt
≤
K
2
h
u
1
t
|
Γ
A
1
t
− h
u
2
t
|
Γ
A
2
2
dxdt ≤
Q
α
ε
t
K
1
f
1
− f
2
dxdt
T
0
K
2
A
1
t
u
1
u
2
h
m
r
drdxdt −→ −
Q
α
ε
t
u
1
− u
2
dxdt,
Q
K
t
K
1
f
1
− f
2
dxdt,
T
0
K
2
h
m
u
1
t
|
Γ
A
1
A
1
t
− A
2
t
α
ε
t
dt.
3.28
20 Boundary Value Problems
As for the second term in 3.26, we have
Q
N
i,j1
sign
0
u
m
u
2
a
ij
x, u
2
D
j
u
2
D
i
u
2
dxdt
Q
N
i,j1
sign
0
−
Q
N
i,j1
sign
0
u
1
− u
2
α
ε
h
m
u
2
a
ij
x, u
2
i,j1
a
ij
x, u
k
D
j
u
k
D
i
u
k
dxdt, k 1, 2. 3.30
As u
1
, u
2
are renormalized solutions, noting 2.7, we prove
lim
m → ∞
J
1
lim
m → ∞
J
2
T
0
u
t
,v
V
,V
dt
Q
N
i,j1
a
ij
x, u
D
j
uD
i
vdxdt
Q
fvdxdt
x, uD
j
u ∈ L
2
Q,i,j 1, 2 N, we have u ∈ L
2
0,T; V
,
hence u ∈ C0,T; L
2
Ω, for any given h ∈ C
1
c
R and ξ ∈ W. Taking v ξhu in 4.1,we
have
T
0
u
t
,ξh
u
dt −
Q
N
h
u
t
|
Γ
ξ
t
|
Γ
dt.
4.3
By 12, Lemma 1.4, we have
T
0
u
t
,ξh
u
i,j1
a
ij
x, u
D
j
uD
i
ξh
u
dxdt
Q
fξh
u
dxdt
T
0
A
0
T
m1
s−T
m
sds,itiseasytoseethat0≤ S
m
r ≤|r|. Taking
v T
m1
u − T
m
u in 4.1,weget
Ω
S
m
u
x, T
dx
Q
N
i,j1
a
ij
m
u
dxdt
T
0
A
t
T
m1
u
t
|
Γ
− T
m
u
t
|
Q
N
i,j1
a
ij
x, u
D
j
uD
i
T
m1
u
− T
m
u
dxdt 0.
4.7
22 Boundary Value Problems
Conversely, assume that u is a renormalized solution. Applying 2.6 with hu
Hn 1 −|u|, where H ∈ C
T
0
A
t
ξ
t
|
Γ
dt.
4.8
Hence u is a weak solution to problem P.
ii Due to u ∈ L
∞
Q, assumption 3.1 and the definition of a renormalized solution
to problem P , ii is an immediate consequence of i.
Remark 4.2. Theorems 2.5 and 3.4 improve those results of 1, 3, 8.
Acknowledgments
This work is supported by NCET of Chinano: 060275 and Foundation of Maths XofDLUT.
References
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13 E. Zeidler, Nonlinear Functional Analysis and Its Applications. Part 2B: Nonlinear Monotone Operators,
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Boundary Value Problems 23
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ˇ
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