Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 81, pp. 1–13.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
REACTION DIFFUSION EQUATIONS WITH BOUNDARY
DEGENERACY
HUASHUI ZHAN
Abstract. In this article, we consider the reaction diffusion equation
∂u
= ∆A(u), (x, t) ∈ Ω × (0, T ),
∂t
with the homogeneous boundary condition. Inspired by the Fichera-Oleǐnik
theory, if the equation is not only strongly degenerate in the interior of Ω, but
also degenerate on the boundary, we show that the solution of the equation is
free from any limitation of the boundary condition.
1. Introduction
Consider the equation
∂u
= ∆A(u), (x, t) ∈ Ω × (0, T ),
(1.1)
∂t
with the homogeneous boundary condition, where Ω ⊂ RN is an open bounded
domain with the appropriately smooth boundary ∂Ω, and
Z u
A(u) =
a(s)ds, a(s) ≥ 0, a(0) = 0.
(1.2)
0
One of particular cases of equation (1.1) is
∂u
= ∆um .
(1.3)
∂t
According to the degenerate parabolic equation theory, if there is no interior point
in the set {s ∈ R : a(s) = 0}, as usual we say that equation (1.1) is weakly
degenerate; otherwise, we say that equation (1.1) is strongly degenerate.
For the Cauchy problem of equation (1.1), Vol’pert and Hudjave [14] investigated
its solvability. Thereafter, much attention has dedicated to the study of its wellposedness [1, 2, 3, 10, 16, 17, 18].
When we consider the initial-boundary value problem of equation (1.1), usually
one needs the initial condition as
u(x, 0) = u0 (x),
x ∈ Ω.
2010 Mathematics Subject Classification. 35L65, 35K85, 35R35.
Key words and phrases. Reaction diffusion equation; Fichera-Oleǐnik theory;
boundary condition; degeneracy.
c
2016
Texas State University.
Submitted May 14, 2015. Published March 23, 2016.
1
(1.4)
2
H. ZHAN
EJDE-2016/81
However, can we impose the Dirichlet homogeneous boundary condition
u(x, t) = 0,
(x, t) ∈ ∂Ω × (0, T ),
(1.5)
into the problem?
Obviously, when both (1.2) and (1.5) hold, equation (1.1) is not only degenerate
in the interior of Ω, but also on the boundary ∂Ω. If it is weakly degenerate, we will
show that equation (1.1) can be imposed by the boundary condition (1.5) actually.
While, if it is in the strongly degenerate case, we will show that the solution of
equation (1.1) is free from any limitation of the boundary condition. Let us give a
brief review on the corresponding problems.
The memoir by Tricomi [13], as well as subsequent investigations of equations of
mixed type, elicited interest in the general study of elliptic equations degenerating
on the boundary of the domain. The paper by Keldyš [9] plays a significant role in
the development of the theory. It was brought to light that in the case of elliptic
equations degenerating on the boundary, under definite assumptions, a portion of
the boundary may be free from the prescription of boundary conditions. Later,
Fichera[6, 7] and Oleǐnik [11, 12] developed the general theory of second order
equations with a nonnegative characteristic form, which, in particular contains
those degenerating assumptions on the boundary. We can call the theory as the
Fichera-Oleǐnik theory.
To study the boundary value problem of a linear degenerate elliptic equation:
N
+1
X
∂u
∂2u
e ⊂ RN +1 ,
+
br (x)
+ c(x)u = f (x), x ∈ Ω
a (x)
∂x
∂x
∂x
r
s
r
r=1
r,s=1
N
+1
X
rs
(1.6)
it needs and only needs the part boundary condition. In detail, let {ns } be the unit
e and denote
inner normal vector of ∂ Ω
e : ars nr ns = 0, (br − ars )nr < 0},
Σ2 = {x ∈ ∂ Ω
xs
(1.7)
e : ars ns nr > 0}.
Σ3 = {x ∈ ∂ Ω
Then, to ensure the well-posedness of equation (1.7), according to the FicheraOleinik theory, the suitable boundary condition is
u|Σ2 ∪Σ3 = g(x).
(1.8)
rs
In particular, if the matrix (a ) is definite positive, (1.8) is the regular Dirichlet
boundary condition.
If A−1 exists, in other words, equation (1.1) is weakly degenerate, let v = A(u)
and u = A−1 (v). Then it has
∆v − (A−1 (v))t = 0.
(1.9)
According to the Fichera-Oleinik theory, one can impose the Dirichlet homogeneous
boundary condition (1.5).
But, if equation (1.1) is strongly degenerate, then A−1 does not exist, we can
not deal with it as equation (1.9). We rewrite equation (1.1) as
∂u
= a(u)∆u + a0 (u)|∇u|2 , (x, t) ∈ Ω × (0, T ),
(1.10)
∂t
and let t = xN +1 . We regard the strongly degenerate parabolic equation (1.10) as
the form of a “linear” degenerate elliptic equation as follows: when i, j = 1, 2, . . . , N ,
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REACTION DIFFUSION EQUATIONS
3
aii (x, t) = a(u(x, t)), aij (x, t) = 0, i 6= j, then it has
ij
a
0
(e
ars )(N +1)×(N +1) =
.
0 0
If a(0) = 0, then equation (1.10) is not only strongly degenerate in the interior of
Ω, but also degenerate on the boundary ∂Ω. We can see that Σ3 is an empty set,
while
(
∂u
0
ebs (x, t) = a (u) ∂xi , 1 ≤ s ≤ N,
−1,
s = N + 1.
Under this observation, according to the Fichera-Oleinik theory, the initial condition (1.4) is always required. But on the lateral boundary ∂Ω × (0, T ), by a(0) = 0,
the part of boundary in which we should give the boundary value is
∂u ∂u − a0 (0)
ni < 0 = ∅,
(1.11)
Σp = x ∈ ∂Ω : a0 (0)
x∈∂Ω
x∈∂Ω
∂xi
∂xi
where {ni } is the unit inner normal vector of ∂Ω. This implies that no any boundary
condition is necessary. In other words, the initial-boundary problem of equation
(1.1) is actually free from the limitation of the boundary condition. Certainly, the
above discussion is based on the assumption that there is a classical solution of
equation (1.1). In fact, due to the strongly degenerate properties of A(u), equation
(1.1) generally only has a weak solution. So it remains to be clarified whether
the solution of the equation is actually free from the limitation of the boundary
condition or not?
2. Main results
For small η > 0, let
Z
Sη (s) =
s
hη (τ )dτ,
hη (s) =
0
|s|
2
(1 − )+ .
η
η
(2.1)
Obviously, hη (s) ∈ C(R), and
hη (s) ≥ 0,
|shη (s)| ≤ 1,
lim Sη (s) = sign s,
η→0
|Sη (s)| ≤ 1,
lim sSη0 (s) = 0.
(2.2)
η→0
Definition 2.1. A function u is said to be the entropy solution of (1.1) with the
initial condition (1.4), if
1. u satisfies
Z up
∂
u ∈ BV (QT ) ∩ L∞ (QT ),
a(s)ds ∈ L2 (QT ).
(2.3)
∂xi 0
2. For any ϕ ∈ C02 (QT ), ϕ ≥ 0, k ∈ R, with a small η > 0, u satisfies
ZZ h
Z up
i
Iη (u − k)ϕt + Aη (u, k)∆ϕ − Sη0 (u − k)|∇
a(s)ds|2 ϕ dx dt ≥ 0. (2.4)
QT
0
3. The initial condition is true in the sense that
Z
lim
|u(x, t) − u0 (x)|dx = 0.
t→0
Ω
(2.5)
4
H. ZHAN
EJDE-2016/81
One can see that if (1.1) has a classical solution u, by multiplying (1.1) by
ϕ1 Sη (u−k) and integrating it over QT , we are able to show that u satisfies Definition
2.1. On the other hand, letting η → 0 in (2.4), we have
ZZ
[|u − k|ϕt + sign(u − k)(A(u) − A(k))∆ϕ] dx dt ≥ 0.
QT
Thus if u is the entropy solution as in Definition 2.1, then u is a entropy solution
as defined in [10, 14] et al.
Theorem 2.2. Suppose that A(s) is C 3 and u0 (x) ∈ L∞ (Ω). Suppose that
A0 (0) = a(0) = 0.
(2.6)
Then (1.1) with the initial condition (1.4) has a entropy solution in the sense of
Definition 2.1.
Theorem 2.3. Suppose that A(s) is C 2 . Let u and v be solutions of (1.1) with the
different initial values u0 (x), v0 (x) ∈ L∞ (Ω) respectively. Suppose that the distance
function d(x) = dist(x, Σ) < λ satisfies
Z
1
dx ≤ c,
(2.7)
|∆d| ≤ c,
λ Ωλ
where λ is a sufficiently small constant, and Ωλ = {x ∈ Ω, d(x, ∂Ω) < λ}. Then
Z
Z
|u(x, t) − v(x, t)|dx ≤
|u0 − v0 |dx + ess sup(x,t)∈∂Ω×(0,T ) |u(x, t) − v(x, t)|.
Ω
Ω
(2.8)
3. Proof of Theorem 2.2
Let Γu be the set of all jump points of u ∈ BV (QT ), v be the normal of Γu at
X = (x, t), u+ (X) and u− (X) be the approximate limit of u at X ∈ Γu with respect
to (v, Y − X) > 0 and (v, Y − X) < 0 respectively. For the continuous function
p(u, x, t) and u ∈ BV (QT ), we define
Z 1
pb(u, x, t) =
p(τ u+ + (1 − τ )u− , x, t)dτ,
(3.1)
0
which is called the composite mean value of p.
t
For a given t, we denote Γtu , H t , (v1t , . . . , vN
) and ut± as all jump points of
t
u(·, t), Housdorff measure of Γu , the unit normal vector of Γtu , and the asymptotic
limit of u(·, t) respectively. Moreover, if f (s) ∈ C 1 (R) and u ∈ BV (QT ), then
f (u) ∈ BV (QT ) and
holds, where xN +1
∂u
∂f (u)
= fb0 (u)
, i = 1, 2, . . . , N, N + 1,
∂xi
∂xi
= t.
(3.2)
Lemma 3.1. Let u be a solution of (1.1). Then
a(s) = 0,
s ∈ I(u+ (x, t), u− (x, t))
a.e. on Γu ,
(3.3)
where I(α, β) denote the closed interval with endpoints α and β, and (3.3) is in the
sense of Hausdorff measure HN (Γu ).
The proof of the above lemma is similar to the one in [18], so we omit it.
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REACTION DIFFUSION EQUATIONS
5
Lemma 3.2 ([4]). Assume that Ω ⊂ RN is an open bounded set and let fk , f ∈
Lq (Ω), as k → ∞, fk * f weakly in Lq (Ω) and 1 ≤ q < ∞. Then
lim inf kfk kqLq (Ω) ≥ kf kqLq (Ω) .
k→∞
(3.4)
We now consider the regularized problem
∂u
= ∆A(u) + ε∆u, (x, t) ∈ Ω × (0, T ),
∂t
with the initial and boundary conditions
u(x, 0) = u0 (x),
u(x, t) = 0,
x ∈ Ω,
(x, t) ∈ ∂Ω × (0, T ).
(3.5)
(3.6)
(3.7)
It is well known that there are classical solutions uε ∈ C 2 (QT ) ∩ C 3 (QT ) of this
problem provided that A(s) satisfies the assumptions in Theorem 2.2. One can
refer to [15] or the eighth chapter of [8] for details.
We need to make some estimates for uε of (3.5). Firstly, since u0 (x) ∈ L∞ (Ω)
is sufficiently smooth, by the maximum principle we have
|uε | ≤ ku0 kL∞ ≤ M.
(3.8)
Secondly, let us make the BV estimates of uε . To the end, we begin with the
local coordinates of the boundary ∂Ω.
Let δ0 > 0 be small enough. Denote
E δ0 = {x ∈ Ω̄; dist(x, Σ) ≤ δ0 } ⊂ ∪nτ=1 Vτ ,
where Vτ is a region, and one can introduce local coordinates of Vτ ,
yk = Fτk (x)
(k = 1, 2, . . . , N ),
yN |Σ = 0,
(3.9)
with Fτk appropriately smooth and FτN = FlN , such that the yN −axes coincides
with the inner normal vector.
Lemma 3.3 ([15]). Let uε be the solution of equation (3.5) with (3.6), (3.7). If
the assumptions of Theorem 2.2 are true, then
Z
∂uε
∂uε
ε
|
|dσ ≤ c1 + c2 (|∇uε |L1 (Ω) + |
|L1 (Ω) ),
(3.10)
∂n
∂t
Σ
with constants ci , i = 1, 2 independent of ε.
We have the following important estimates of the solutions uε of equation (3.5)
with (3.6), (3.7).
Theorem 3.4. Let uε be the solution of equation (3.5) with (3.6), (3.7). If the
assumptions of Theorem 2.2 are true, then
| grad uε |L1 (Ω) ≤ c,
where | grad u|2 =
PN
∂u 2
i=1 | ∂xi |
(3.11)
2
+ | ∂u
∂t | , and c is independent of ε.
Proof. Differentiate (3.5) with respect to xs , s = 1, 2, ·, N, N + 1, xN +1 = t, and
S (| grad u |)
sum up for s after multiplying the resulting relation by uεxs η| grad uε |ε . In what
follows, we simply denote uε by u, denote ∂Ω by Σ, and denote dσ by the surface
integral unite on Σ.
6
H. ZHAN
EJDE-2016/81
Integrating it over Ω yields
Z
Z | grad u|
Z
Z
d
∂uxs
Sη (| grad u|)
∂
Sη (τ )dτ dx =
uxs
dx =
Iη (| grad u|dx,
| grad u|
dt Ω
Ω ∂t
Ω ∂t 0
where pairs of the indices of s imply a summation from 1 to N + 1, pairs of the
indices of i, j imply a summation from 1 to N , and {ni }N
i=1 is the inner normal
vector of Ω. So we have
Z
Sη (| grad u|)
∆(a(u)uxs )uxs
dx
| grad u|
Ω
Z
Sη (| grad u|)
∂ 0
[a (u)uxi uxs + a(u)uxi xs ]uxs
dx
=
∂x
| grad u|
i
(3.12)
ZΩ
∂
Sη (| grad u|)
0
=
dx
(a (u)uxi uxs )uxs
| grad u|
Ω ∂xi
Z
∂
Sη (| grad u|)
+
(a(u)uxi xs )uxs
dx ,
∂x
| grad u|
i
Ω
Z
∂
Sη (| grad u|)
dx
(a0 (u)uxi uxs )uxs
∂x
| grad u|
i
Ω
N
+1 Z
X
∂
Sη (| grad u|)
dx
=
(a0 (u)uxi )u2xs
∂x
| grad u|
i
s=1 Ω
Z
∂
+
a0 (u)uxi
Iη (| grad u|)dx
∂xi
Z Ω
∂
(3.13)
=
(a0 (u)uxi )| grad u|Sη (| grad u|)dx
∂x
Ω
Z i
Z
∂
(a0 (u)uxi )dx
−
a0 (u)uxi ni Iη (| grad u|)dσ −
Iη (| grad u|)
∂xi
Σ
Ω
Z
∂
=
(a0 (u)uxi ) [| grad u|Sη (| grad u|) − Iη (| grad u|)] dx
Ω ∂xi
Z
−
a0 (u)uxi ni Iη (| grad u|)dσ ,
Σ
Z
Sη (| grad u|)
∂
(a(u)uxi xs )uxs
dx
∂x
| grad u|
i
Ω
Z
∂
∂
=
(a(u)uxi xs )
Iη (| grad u|)dx
∂ξs
Ω ∂xi
Z
(3.14)
∂
=−
a(u)uxi xs ni
Iη (| grad u|)dσ
∂ξs
Σ
Z
2
∂ Iη (| grad u|)
−
a(u)
uxs xi uxp xi dx ,
∂ξs ∂ξp
Ω
where ξs = uxs .
Z
Sη (| grad u|)
ε
∆uxs uxs
dx
| grad u|
Ω
Z
Z 2
∂Iη (| grad u|)
∂ Iη (| grad u|)
= −ε
ni dσ − ε
uxs xi uxp xi dx.
∂x
∂ξs ∂ξp
i
Σ
Ω
(3.15)
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7
From (3.12)–(3.15), by the assumption a(0) = 0, we have
Z
d
Iη (| grad u|dx
dt Ω
Z
∂
=
(a0 (u)uxi ) [| grad u|Sη (| grad u|) − Iη (| grad u|)] dx
Ω ∂xi
Z
∂ 2 Iη (| grad u|)
−
uxs xi uxp xi dx
a(u)
∂ξs ∂ξp
Ω
Z 2
∂ Iη (| grad u|)
−ε
uxs xi uxp xi dx
∂ξs ∂ξp
Ω
Z
hZ
i
∂Iη (| grad u|)
−
ni dσ .
a0 (u)uxi ni Iη (| grad u|)dσ + ε
∂xi
Σ
Σ
Note that on Σ, we have
0 = ε∆u + ∆A(u),
u = 0,
(3.16)
(3.17)
then the surface integrals in (3.16) can be rewritten as
Z
i
h Z ∂I (| grad u|)
η
ni dσ +
a0 (u)uxi ni Iη (| grad u|)dσ
S=− ε
∂xi
Σ
Z Σ
∂Iη (| grad u|)
Iη (| grad u|) = −ε
ni − ∆u
dσ
∂u
∂xi
Σ
∂n
Z
∂Iη (| grad u|)
Iη (| grad u|) ni − ∆u
dσ
+
a(u)
∂u
∂xi
Σ
∂n
Z
∂Iη (| grad u|)
Iη (| grad u|) = −ε
dσ.
ni − ∆u
∂u
∂x
i
Σ
∂n
Since uxN +1 |Σ = ut |Σ = 0, we have
Z
∂u
lim S = −ε
sign( )(uxi xj nj ni − ∆u)dσ.
η→0
∂n
Σ
Using the local coordinates on Vτ , τ = 1, 2, . . . , n, we have
yk = Fτk (x),
k = 1, 2, . . . , N,
(3.18)
ym |Σ = 0.
By a direct computation (refer to [15]), on Σ ∩ Vτ we obtain
uxi xj =
N
X
uyN yk FxNi Fxkj +
N
−1
X
k=1
k=1
PN
uxi xj nj ni =
uyN yk FxNi Fxkj + uym Fxmi xj ,
N k
N N
k=1 uyN yk Fxi Fxj Fxj Fxi
| grad F N |2
+
N
−1
X
uyN yk Fxki FxNj +
uym Fxmi xj FxNj FxNi
k=1
| grad F N |2
in which F k = Fτk . since the inner normal vector is
~n = −(
∂F N
∂F N
,...,
) = − grad F N ,
∂x1
∂xN
it follows that
uxi xj nj ni − ∆u = uym
Fm FNFN
xi xj xj xi
| grad F N |2
− Fxmi xi .
By Lemma 3.3, we see that limη→0 S can be estimated by | grad u|L1 (Ω) .
,
8
H. ZHAN
EJDE-2016/81
By letting η → 0, from
lim [| grad u|Sη (| grad u|) − Iη (| grad u|)] = 0,
η→0
we have
Z
Z
d
| grad u|dx ≤ c1 + c2
| grad u|dx.
dt Ω
Ω
Further, by Gronwall’s Lemma we have
Z
| grad u|dx ≤ c.
(3.19)
Ω
By (3.5) and (3.19), it is easy to show that
ZZ
(a(uε ) + ε)|∇uε |2 dx dt ≤ c.
(3.20)
QT
Thus, there exists a subsequence {uεn } of uε and a function u ∈ BV (QT )
∩L∞ (QT ) such that uεn → u a.e. on QT .
Proof. We now prove that u is a generalized solution of equation (1.1) with the
initial condition (1.4). For any ϕ(x, t) ∈ C01 (QT ), we have
ZZ
Z uε p
Z up
∂
∂
[
a(s)ds −
a(s)ds]ϕ(x, t) dx dt
∂xi 0
QT ∂xi 0
Z up
Z Z h Z uε p
i
a(s)ds −
a(s)ds ϕxi (x, t) dx dt.
=−
QT
0
0
By a limiting process, we know the above equality is also true for any ϕ(x, t) ∈
L2 (QT ). By (3.20), we have
Z uε p
Z up
∂
∂
a(s)ds *
a(s)ds
∂xi 0
∂xi 0
weakly in L2 (QT ) for i = 1, 2, . . . , N . This implies
Z up
∂
a(s)ds ∈ L2 (QT ), i = 1, 2, . . . , N.
∂xi 0
Thus u satisfies (2.3) in Definition 2.1.
Let ϕ ∈ C02 (QT ), ϕ ≥ 0, and {ni } be the inner normal vector of Ω. Multiplying
(3.5) by ϕSη (uε − k), and integrating it over QT , we obtain
ZZ
ZZ
Iη (uε − k)ϕt dx dt +
Aη (uε , k)∆ϕ dx dt
QT
QT
ZZ
ZZ
−ε
∇uε · ∇ϕSη (uε − k) dx dt − ε
|∇uε |2 Sη0 (uε − k)ϕ dx dt (3.21)
Q
QT
ZZ T
2 0
−
a(uε )|∇uε | Sη (uε − k)ϕ dx dt = 0.
QT
By Lemma 3.2,
ZZ
Sη0 (uε − k)a(uε )
lim inf
ε→0
QT
ZZ
Sη0 (u − k)|∇
≥
QT
Z
0
∂uε ∂uε
ϕ dx dt
∂xi ∂xi
u
p
a(s)ds|2 ϕ dx dt.
(3.22)
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9
Let ε → 0 in (3.21). By (3.22), we get (2.4). Finally, we can prove equality (2.5)
in a similar manner as that in [14] or [18], we omit the details.
4. Proof of Theorem 2.3
Proof. Let u and v be two entropy solutions of (1.1) in the sense of Definition 2.1.
Suppose the initial values are
u(x, 0) = u0 (x),
v(x, 0) = v0 (x) .
(4.1)
C02 (QT ),
By Definition 2.1, for any ϕ ∈
ϕ ≥ 0, and η > 0, k, l ∈ R, we have
Z up
ZZ h
i
Iη (u − k)ϕt + Aη (u, k)∆ϕ − Sη0 (u − k)|∇
a(s)ds|2 ϕ dx dt ≥ 0, (4.2)
QT
ZZ
h
Iη (v − l)ϕτ + Aη (v, l)∆ϕ − Sη0 (v − l)|∇
Z
0
v
i
p
a(s)ds|2 ϕ dy dτ ≥ 0. (4.3)
0
QT
Let ψ(x, t, y, τ ) = φ(x, t)jh (x − y, t − τ ), where φ(x, t) ≥ 0, φ(x, t) ∈ C0∞ (QT ), and
jh (x − y, t − τ ) = ωh (t − τ )ΠN
i=1 ωh (xi − yi ),
1 s
ωh (s) = ω( ), ω(s) ∈ C0∞ (R), ω(s) ≥ 0, ω(s) = 0
h h
Z
∞
if |s| > 1,
(4.4)
(4.5)
ω(s)ds = 1.
−∞
We choose k = v(y, τ ), l = u(x, t), ϕ1 = ψ(x, t, y, τ ) in (4.2)-(4.3), integrating
over QT we obtain
ZZ ZZ
[Iη (u − v)(ψt + ψτ ) + Aη (u, v)∆x ψ + Aη (v, u)∆y ψ]
QT
QT
(4.6)
Z vp
Z up
− Sη0 (u − v) |∇
a(s)ds|2 + |∇
a(s)ds|2 ψ dx dt dy dτ = 0.
0
0
Clearly,
∂jh
∂jh
∂jh
+
= 0,
+
∂t
∂τ
∂xi
∂ψ ∂ψ
∂φ
+
=
jh ,
∂t
∂τ
∂t
∂jh
= 0, i = 1, . . . , N ;
∂yi
∂ψ
∂ψ
∂φ
+
=
jh .
∂xi
∂yi
∂xi
For the third and the fourth terms in (4.6), we have
ZZ
[Aη (u, v)∆x ψ + Aη (v, u)∆y ψ] dx dt dy dτ
QT
ZZ
ZZ
=
{Aη (u, v)(∆x φjh + 2φxi jhxi + φ∆jh ) + Aη (v, u)φ∆y jh } dx dt dy dτ
Q
Q
ZZ T ZZ T
=
{Aη (u, v)∆x φjh + Aη (u, v)φxi jhxi + Aη (v, u)φxi jhyi } dx dt dy dτ
QT
QT
ZZ
ZZ
−
QT
∂u
\
−
{a(u)S
η (u − v)
∂x
i
QT
Z
v
u
\
∂u
a(s)Sη0 (s − v)ds
)φjhxi } dx dt dy dτ,
∂xi
where
Z
\
a(u)S
η (u − v) =
0
1
a(su+ + (1 − s)u− )Sη (su+ + (1 − s)u− − v)ds,
10
H. ZHAN
v
Z
0
\
a(s)S
η (s − v)ds =
u
1
Z
Z
Q
Z
Sη0 (u − v) |∇x
Q
T
ZZ
a(σ)Sη (σ − su+ − (1 − s)u− )dσds.
u
p
a(s)ds|2 + |∇y
Sη0 (u
=
Sη0 (u − v)∇x
+2
u
QT
ZZ
1
Z
QT
0
a(s)ds|
2
ψ dx dt dy dτ
0
v
a(s)ds · ∇y
p
a(s)dsψ dx dt dy dτ .
0
p
a(σ)Sη0 (σ − δ) dσ dδψ dx dt dy dτ
δ
1
Z
p
=
QT
p
v
Z
p
a(δ)
v
QT
p
Z
0
By Lemma 3.1, we have
ZZ ZZ
Z
∇x ∇y
v
a(s)ds| − |∇y
u
Z
QT
ZZ
Z
0
ZZ
QT
p
− v) |∇x
QT
ZZ
p
a(s)ds|2 ψ dx dt dy dτ
0
u
Z
v
Z
0
T
ZZ
QT
v
su+ +(1−s)u−
0
Since
ZZ ZZ
EJDE-2016/81
p
a(su+ + (1 − s)u− ) a(σv + + (1 − σ)v −
0
× ×Sη0 [σv + + (1 − σ)v − − su+ − (1 − s)u− ] d dσ∇x u∇y v dx dt dy dτ
ZZ ZZ Z 1 Z 1
=
Sη0 [σv + + (1 − σ)v − − su+ − (1 − s)u− ] d dσ
QT
QT
0
0
p
p
\
\
× a(u)∇x u a(v)∇y v dx dt dy dτ
ZZ ZZ Z 1 Z 1
Z up
Z
=
Sη0 (v − u)∇x
a(s)ds∇y
QT
QT
0
0
0
v
p
a(s)ds dx dt dy dτ.
0
and
ZZ
ZZ
u
Z
p
∇x ∇y
QT
QT
ZZ
ZZ
×
QT
p
a(σ)Sη0 (σ − δ) dσ dδψ dx dt dy dτ
1
p
=
Q
Z Tv
v
δ
v
Z
Z
a(δ)
a(su+ + (1 − s)u− )
0
p
∂u
a(σ)Sη0 (σ − su+ − (1 − s)u− )dσds
jhxi φ dx dt dy dτ.
∂x
+
−
i
su +(1−s)u
We further have
Z v
ZZ ZZ
\
∂u
∂u
\
−
a(s)Sη0 (s − u)ds
)jhxi φ dx dt dy dτ
(a(u)Sη (u − v)
∂x
∂x
i
i
u
QT
Q
Z vp
ZZ T ZZ
Z up
+2
Sη0 (u − v)∇x
a(s)ds · ∇y
a(s)dsψ dx dt dy dτ
ZZ
QT
QT
ZZ
hZ
=
QT
Z 1
Z
QT
v
0
a(su+ + (1 − s)u− )Sη (su+ + (1 − s)u− − v)ds
0
−
su+ +(1−s)u−
0
Z
+2
0
a(σ)Sη0 (σ − su+ − (1 − s)u− )dσds
1
p
0
1
a(su+ + (1 − s)u− )
Z
v
su+ +(1−s)u−
p
a(σ)Sη0 (σ − su+
EJDE-2016/81
REACTION DIFFUSION EQUATIONS
11
i ∂u
− (1 − s)u− )dσds
jhxi φ dx dt dy dτ
∂xi
ZZ ZZ Z 1 Z v
p
p
[ a(σ) − a(su+ + (1 − s)u− )]
=−
QT
QT
0
su+ +(1−s)u−
× Sη0 (σ − su+ − (1 − s)u− )dσds
∂u
jhxi φ dx dt dy dτ → 0,
∂xi
as η → 0. Since
lim Aη (u, v) = lim Aη (v, u) = sign(u − v)[A(u) − A(v)],
η→0
η→0
we have
lim [Aη (u, v)φxi jhxi + Aη (u, v)φyi jhyi ] = 0.
η→0
By (4.6)–(4.7) and letting η → 0, h → 0 in (4.6), we obtain
ZZ
[|u(x, t) − v(x, t)|φt + |A(u) − A(v)|∆φ] dx dt ≥ 0.
(4.7)
(4.8)
QT
Let δε be the mollifier. For any given ε > 0, y = (y1 , . . . , yN ), δε (y) is defined by
1 y
δε (y) = N δ( ),
ε
ε
where
(
1
1 |y|2 −1
e
, if |y| < 1,
A
δ(y) =
0,
if |y| ≥ 1,
with
Z
1
e |y|2 −1 dx.
A=
B1 (0)
Especially, we can choose φ in (4.8) by
φ(x, t) = ωλε (x)η(t),
C0∞ (0, T ),
where η(t) ∈
and ωλε (x) is the mollified function of ωλ . Let ωλ (x) ∈
C02 (Ω) be defined as follows: for any given small enough 0 < λ, 0 ≤ ωλ ≤ 1,
ω|∂Ω = 0 and
ωλ (x) = 1, if d(x) = dist(x, ∂Ω) ≥ λ,
where 0 ≤ d(x) ≤ λ and
ωλ (d(x)) = 1 −
(d(x) − λ)2
.
λ2
Then ωλε = ωλ ∗ δε (d),
0
ωλε
(d) = −
Z
ωλ0 (d − s)δε (s)ds
{|s|<ε}∩{0<d−s<λ}
Z
=−
{|s|<ε}∩{0<d−s<λ}
2(d − s − λ)
δε (s)ds.
λ2
We know that
∆φ = η(t)∆(ωλε (d(x)))
0
= η(t)∇(ωλε
(d)∇d)
00
0
= η(t)[ωλε
(d)|∇d|2 + ωλε
(d)∆d]
12
H. ZHAN
= η(t)[−
2
λ2
EJDE-2016/81
Z
0
ds + ωλε
(d)∆d].
{|s|<ε}∩{0<d−s<λ}
By using condition (2.7), and that |∇d(x)| = 1, a.e. x ∈ Ω, from (4.8) we have
Z
Z TZ
0
|u(x, t) − v(x, t)|φt dx dt + c
η(t)|ωλε
(d)||u − v| dx dt ≥ 0.
(4.9)
QT
0
Ωλ
0
where Ωλ = {x ∈ Ω, d(x, ∂Ω) < λ}. Since |ωλε
(d)| ≤ λc , let ε → 0 in (4.9). We have
Z TZ
Z Z
c T
0
η(t)|ωλε (d)||u − v| dx dt ≤
η(t)|u − v| dx dt.
lim
ε→0 0
λ 0 Ωλ
Ωλ
According to the the definition of the trace of the BV functions [4], we let ε → 0
and λ → 0. Then we have
Z
c ess sup∂Ω×(0,T ) |u(x, t) − v(x, t)| +
|u(x, t) − v(x, t)|ηt0 dx dt ≥ 0.
(4.10)
QT
Let 0 < s < τ < T , and
Z
s−t
η(t) =
αε (σ)dσ,
ε < min{τ, T − s}.
τ −t
Then it follows that
Z
c ess sup∂Ω×(0,T ) |u(x, t) − v(x, t)| +
T
[αε (t − s) − αε (t − τ )]|u − v|L1 (Ω) dt ≥ 0.
0
By letting ε → 0, we obtain
|u(x, τ ) − v(x, τ )|L1 (Ω) ≤ |u(x, s) − v(x, s)|L1 (Ω) + c ess sup∂Ω×(0,T ) |u(x, t) − v(x, t)|.
Consequently, the desired result follows by letting s → 0.
Acknowledgments. This work was supported by NSF of China 11371297 and
NSF of Fujian province in China 2015J01592.
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E-mail address: 2012111007@xmut.edu.cn
Huashui Zhan
School of Applied Mathematics, Xiamen University of Technology, Xiamen, Fujian
361024, China