RESEA R C H Open Access
Local existence and uniqueness of solutions
of a degenerate parabolic system
Dazhi Zhang, Jiebao Sun
*
and Boying Wu
* Correspondence: sunjiebao@126.
com
Department of Mathematics,
Harbin Institute of Technology,
Harbin 150001, PR China,
Abstract
This article deals with a degenerate parabolic system coupled with general nonlinear
terms. Using the method of regularization and monotone iteration technique, we
obtain the local existence of solutions to the Dirichlet initial boundary value problem.
We also establish the uniqueness of the solution if the reaction terms satisfy the
Lipschitz condition.
Keywords: Existence, Uniqueness, Degenerate, Monotone iteration
1 Introduction
In this article, we consider the following degenerate parabolic system
∂u
i
∂t
= u
m
i
i
+ f
i
(x, t, u
1

, x ∈ 
,
(1:3)
where m
i
>1,i=1,2,Q
T
= Ω ×(0,T), Ω is a bounded domain in ℝ
N
with smooth
boundary,
f
i
(
x, t, u
1
, u
2
)
∈ C
(
¯
 × [0, T] ×
R
2
)
and
0 ≤ u
i0
∈ L

obtained as the limit of the solutions of suc h problems. Executing this program o ne
encounters two difficulties. The first is proving that the approximating problems which
are nondegenerate admits a solution, the second difficulty is to est ablish uniform
estimates for these solutions. At last, we establish the uniqueness results when the
reaction terms satisfy the Lipschitz condition.
Since the system (1.1) is degenerate whenever u
1
,u
2
vanish, there is no classical
solution in general. So we focus our main efforts on the discussion of weak solutions
in the sense of the following.
Definition 1.1. A nonnegative vector-valued function u =(u
1
,u
2
)iscalledtobea
weak solution of the problem (1.1)-(1.3) provided that
u
m
i
i
∈ L
2
(0, T; H
1
0
()) ∩ L
∞
(Q



u
i0
(x)ϕ
i
(x,0)dx =

Q
T
f
i
(x, t, u
1
, u
2
)ϕ
i
dxdt
,
for any test function
ϕ
i
∈ C
2
(
¯
Q
T
)

i
(x, t)dx −


u
i0
(x)ϕ
i
(x,0)d
x
=
t

0


f
i
(x, t, u
1
, u
2
)ϕ
i
dxdt,a.e.t ∈ (0, T).
Definition 1.2.Afunctionf = f(u
1
,u
2
) is said to be quasimonotone nondecreasing

1
(ℝ) such that


f
i
(x, t, u
1
, u
2
)


≤ min

g(u
1
), g(u
2
)

for all (x, t) ∈ Q
T
, u
1
, u
2
∈ R
.
2 Existence and uniqueness

)+f
iε
(x, t, u
1
, u
2
), (x, t) ∈ Q
T
,
(2:1)
u
i
(
x, t
)
=0,
(
x, t
)
∈ ∂ ×
(
0, T
),
(2:2)
Zhang et al. Advances in Difference Equations 2011, 2011:12
/>Page 2 of 11
u
i
(
x,0

× R
2
,andf
iε
satisfies the assumptions (A0), (A1),
u
i0ε
(x) ∈ C
∞
0
(
)
,
u
m
i
i0
ε
→ u
m
i
i0
,
u
m
i
i0
ε
→ u
m

(k)
i
)
m
i
−1
+ ε)∇u
(k)
i
)=f
iε
(x, t, u
(k−1)
1ε
, u
(k−1)
2ε
), (x, t) ∈ Q
T
,
(2:4)
u
(k)
i
ε
(x, t)=0, (x, t) ∈ ∂ × (0, T)
,
(2:5)
u
(k)

(k
−1
)
1
ε
, u
(k
−1
)
2
ε
)
is smooth. The choice of the initial iteration value which will be
obtained by the quasimonotone property of (f
1
,f
2
) would be cruc ial to ensure that the
above sequence converges to a solution of the generalized problem.
Let
(u
−
(0)
1
ε
(x, t), u
−
(0)
2
ε

i
(u
−
(1)
i
)
m
i
−1
+ ε)∇u
−
(1)
i
)=f
iε
(x, t, u
−
(0)
1ε
, u
−
(0)
2ε
), (x, t) ∈ Q
T
,
u
−
(1)
iε

−
(0)
2
ε
.
Then the quasimonotone nondecreasing property of f
iε
shows that
f
1ε
(x, t, u
−
(1)
1ε
, u
−
(1)
2ε
) ≥ f
1ε
(x, t, u
−
(0)
1ε
, u
−
(1)
2ε
) ≥ f
1ε

−
(0)
2
ε
) ≥ f
2ε
(x, t, u
−
(0)
1
ε
, u
−
(0)
2
ε
)
.
Then we can also obtain a classical solution
(u
−
(
2
)
1
ε
, u
−
(
2

(0)
iε
≤ u
−
(1)
iε
≤ u
−
(2)
iε
≤ ···≤ u
−
(k)
iε
≤ ···
.
With the similar method, by setting
(
¯
u
(0)
1ε
(x, t),
¯
u
(0)
2ε
(x, t)) = (sup
Q
T

/>Page 3 of 11
∂
¯
u
(
1
)
i
∂t
− div((m
i
(
¯
u
(1)
i
)
m
i
−1
+ ε)∇
¯
u
(1)
i
)=f
iε
(x, t,
¯
u

u
(1)
1
ε
≤
¯
u
(0)
1
ε
,
¯
u
(1)
2
ε
≤
¯
u
(0)
2
ε
.
And the quasimonotone nondecreasing property of f
iε
also shows that
¯
u
(0)
iε

≤···≤u
−
(k)
iε
≤ u
−
(k+1)
iε
≤
¯
u
(k+1)
iε
≤
¯
u
(k)
iε
≤ ···≤
¯
u
(2)
iε
≤
¯
u
(1)
iε
≤
¯

+1
)
iε
≤ u
−
(k
+1
)
iε
. Since f
iε
is quasimonotone nondecreasing, we have
f
1ε
(x, t, u
−
(k)
1ε
, u
−
(k)
2ε
) ≤ f
1ε
(x, t,
¯
u
(k)
1ε
, u

2ε
(x, t, u
−
(k)
1
ε
,
¯
u
(k)
2ε
) ≤ f
2ε
(x, t,
¯
u
(k)
1ε
,
¯
u
(k)
2ε
)
.
From the iteration equations
∂u
−
(k+1)
i

u
(k+1)
i
∂t
− div((m
i
(
¯
u
(k+1)
i
)
m
1
−1
+ ε)∇
¯
u
(k+1)
i
)=f
iε
(x, t,
¯
u
(k)
1ε
,
¯
u

−
(k
+1
)
iε
≤ u
−
(k
+1
)
iε
. Further we can obtain (2.7).
Let
(u
(k)
1ε
, u
(k)
2ε
)=(u
−
(k)
1
ε
, u
−
(k)
2
ε
)

iε
(i = 1, 2) also shows that
lim
k
→
∞
f
iε
(x, t, u
(k)
1ε
, u
(k)
2ε
)=f
iε
(x, t, u
1ε
, u
2ε
), a.e. in Q
T
.
(2:9)
Therefore, we claim that there exist T
1
Î (0,T] and a positive constant M (indepen-
dent of ε and k), such that for all k,
|
u

i
(0) = ±|u
i0
|
L
∞
()
, i =1,2
.
The results in [16] show that there exists
T
∗
i
∈ (0, T
)
, i =1,2,suchthat
v
±
i
(t
)
exists
on
[0, T
∗
i
]
with
T
∗

Then by setting
T
1
=
1
2
min{T
∗
1
, T
∗
2
}
and
M =max{v
+
i
(T
1
), −v
−
i
(T
1
)
}
, we obtain (2.10).
Nowweshowthat
(u
(k)

ε
)
m
i
t
 (u
m
i
i
ε
)
t
, u
(k)
i
ε
t
 u
iε
t
in
L
2
(Q
T
1
)
as k ® ∞,where⇀ stands for weak
convergence.
Multiplying (2.4) by

(k)
iε

∂u
(k)
iε
∂t
dt dx +

Q
T
1



∇(u
(k)
iε
)
m
i
+ ε∇u
(k)
iε



2
dxdt
=

1



∇(u
(k)
iε
)
m
i
+ ε∇u
(k)
iε



2
dxdt
=

Q
T
1
f
iε
(x, t, u
(k−1)
1ε
, u
(k−1)

m
i
+1
−

u
(k)
iε
(x,0)

m
i
+1

d
x
−
1
2




u
(k)
iε
(x, T
1
)





2
dxdt ≤ C
,
(2:11)
where C is a constant independent of k, ε.
Multiplying (2.4) by
∂
∂
t

u
(k)
iε

m
i
+ εu
(k)
iε

and i ntegrating over
Q
T
1
,byYoung’ s
inequality we have
Zhang et al. Advances in Difference Equations 2011, 2011:12

iε
∂t





2
dxdt
= −
1
2
T
1

0
∂
∂t





∇(u
(k)
iε
)
m
i
+ ε∇u

f
iε
(x, t, u
(k−1)
1ε
, u
(k−1)
2ε
)
∂

u
(k)
iε

m
i
∂t
dxdt
= −
1
2
T
1

0
∂
∂t



∂u
(k)
iε
∂t
dxdt
+
2m
i
m
i
+1

Q
T
1
f
iε
(x, t, u
(k−1)
1ε
, u
(k−1)
2ε
)

u
(k)
iε

(m

(k)
i0ε



2
dx −
1
2





∇

u
(k)
iε
(x, T
1
)

m
i
+ ε∇u
(k)
iε
(x, T
1

f
2
iε
(x, t, u
(k−1)
1ε
, u
(k−1)
2ε
)



u
(k)
iε



m
i
−1
dxd
t
+
2m
i
(m
i
+1)






∂u
(k)
iε
∂t





2
dxdt.
Noticing that the first term of the left side of the above inequality can be rewritten as

Q
T
1
∂u
(k)
iε
∂t
∂(u
(k)
iε
)
m


2
dxdt
.
Then we have
2m
i
(m
i
+1)
2

Q
T
1




∂
∂t

u
(k)
iε

(m
i
+1)/2







∇

u
(k)
i0ε

m
i
+ ε∇u
(k)
i0ε



2
dx +
1
4

Q
T
1
f
2
iε

(k)
iε



m
i
−1
dxdt.
Therefore

Q
T
1




∂
∂t

u
(k)
iε

(m
i
+1)/2



dxdt =
4m
i
(m
i
+1)
2

Q
T
1
(u
(k)
iε
)
m
i
−1




∂
∂t

u
(k)
iε

(m

that, as k ® ∞,
u
(k)
iε
→ u
iε
, f
iε
(x, t, u
(k)
1ε
, u
(k)
2ε
) → f
iε
(x, t, u
1ε
, u
2ε
), a.e. in Q
T
1
,
(2:13)
∂u
(k)
iε
∂t


2
(Q
T
1
)
, s = 1, , n, such that
∂

(u
(k)
iε
)
m
i
+ εu
(k)
iε

∂x
s
ν
s
a.e. in L
2
(Q
T
1
)
.
Hence,

)ϕ
i
dxdt
,
(2:15)
where ν =(ν
1
, , ν
n
),
ϕ
i
∈ C
2
(
¯
Q
T
1
)
with
ϕ
i
|
∂×
(
0,T
1
)
=0

1
ν∇ϕ
i
dxdt,ask →∞
.
(2:16)
For any
w ∈ L
2
(0, T
1
; H
1
0
()
)
,
ζ
∈ C
1
(
¯
Q
T
1
)
,0≤ ζ ≤ 1,
ζ |
∂×
(




∇

u
(k)
iε

m
i
+ εu
(k)
iε




2
dxdt
=

Q
T
1
ζ

u
(k)
iε

m
i
+1
+
ε
2
(u
(k)
i0ε
)
2

dx
+

Q
T
1

1
m
i
+1
(u
(k)
iε
)
m
i
+1

(u
(k)
iε
)
m
i
+ εu
(k)
iε

∇ζ dxdt
.
(2:17)
Zhang et al. Advances in Difference Equations 2011, 2011:12
/>Page 7 of 11
Notice that

Q
T
1
ζ



∇

(u
(k)
iε
)

Q
T
1
ζ ∇w∇

(u
(k)
iε
)
m
i
+ εu
(k)
iε
− w

dxd
t
=

Q
T
1
ζ ∇

(u
(k)
iε
)
m


m
i
+ εu
(k)
iε

f
i
(x, t, u
(k−1)
1ε
, u
(k−1)
2ε
)dxdt
+


ζ (x,0)

1
m
i
+1
(u
(k)
i0ε
)
m

1
2
(u
(k)
iε
)
2

ζ
t
dxdt
−

Q
T
1

u
(k)
iε

m
i
+ εu
(k)
iε

∇

u

Q
T
1
ζ ∇w∇

u
(k)
iε

m
i
+ εu
(k)
iε
− w

dxdt ≥ 0
.
Letting k ® ∞, then

Q
T
1
ζ ((u
iε
)
m
i
+ εu
iε

+

Q
T
1

u
m
i
+1
iε
m
i
+1
+
1
2
u
2
iε

ζ
t
dxdt −

Q
T
1
(u
m

iε
+ εu
iε
)
in (2.15), we obtain

Q
T
1
ζ (u
m
i
iε
+ εu
iε
)f
i
(x, t, u
1ε
, u
2ε
)dxdt
+


ζ (x,0)

u
m
i

2

ζ
t
dxd
t
=

Q
T
1
(u
m
i
iε
+ εu
iε
)ν ∇ζ dxdt +

Q
T
1
ζν∇(u
m
i
iε
+ εu
iε
)dxdt.
Zhang et al. Advances in Difference Equations 2011, 2011:12

m
i
iε
+ εu
iε
))∇ϕ
i
dxdt ≥ 0
,
where
ϕ
i
∈ C
1
(
¯
Q
T
1
)
with
ϕ
i
|
∂×
(
0,T
1
)
=0

, we know that there are function s
u
m
i
i
∈ L
2
(0, T
1
; H
1
0
()
)
,
u
it
, u
m
i
it
∈ L
2
(Q
T
1
)
, i =1,
2, such that for some sub sequence of (u
1ε

∂t
,
∂u
m
i
iε
∂t

∂u
m
i
i
∂t
,inL
2
(Q
T
1
).
Then a similar argument as above shows that u =(u
1
,u
2
) is a weak solution of (1.1)-
(1.3). □
The following is the uniqueness result to the solution of the system.
Theorem 2.2. Assume that f =(f
1
,f
2

∇ϕ
i
dxdt +


u
i
(x, t)ϕ
i
(x, t)dx −


u
i0
(x)ϕ
i
(x,0)d
x
=
t

0


f
i
(x, t, u
1
, u
2

v
i0
(x)ϕ
i
(x,0)d
x
=
t

0


f
i
(x, t, v
1
, v
2
)ϕ
i
dxdt,a.e.t ∈ (0, T).
(2:21)
Zhang et al. Advances in Difference Equations 2011, 2011:12
/>Page 9 of 11
Subtracting the two equations, we get


(u
i
(x, t) − v

) − f
i
(x, t, v
1
, v
2
))ϕ
i
dxds
,
(2:22)
where
(x, s) ≡
1

0
m
i
(θu
i
+(1− θ)v
i
)
m
1
−1
dθ
.
Since (u
1

− v
1
| + |u
2
− v
2
|dxds, i =1,2
.
where C>0 is a bounded constant. Further, we have


|u
1
(x, t) − v
1
(x, t)| + |u
2
(x, t) − v
2
(x, t)|dx ≤ C
t

0


|u
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6. Romanvoskii, YM, Stepanova, NV, Chernavskii, DS: Mathematical biophysics. Nauka, Moscow (1984). (in Russian)
7. Constantin, A, Escher, J, Yin, Z: Global solutions for quasilinear parabolic system. J Differ Equ. 197(1), 73–84 (2004).
doi:10.1016/S0022-0396(03)00165-7
8. Dickstein, F, Escobedo, M: A maximum principle for semilinear parabolic systems and application. Nonlinear Anal. 45(7),
825–837 (2001). doi:10.1016/S0362-546X(99)00419-8
9. Pierre, M, Schmidt, D: Blowup in reaction diffusion systems with dissipation of mass. SIAM J Math Anal. 28(2), 259–269
(1997). doi:10.1137/S0036141095295437
Zhang et al. Advances in Difference Equations 2011, 2011:12
/>Page 10 of 11
10. Wang, MX, Wei, YF: Blow-up properties for a degenerate parabolic system with nonlinear localized sources. J Math Anal
Appl. 343(2), 621–635 (2008). doi:10.1016/j.jmaa.2008.01.073
11. Cui, ZJ, Yang, ZD: Boundedness of global solutions for a nonlinear degenerate parabolic (porous medium) system with
localized sources. Appl Math Comput. 198(2), 882–895 (2008). doi:10.1016/j.amc.2007.09.037
12. Litcanu, G, Morales-Rodrigo, C: Global solutions and asymptotic behavior for a parabolic degenerate coupled system
arising from biology. Nonlinear Anal. 72(1), 77–98 (2008)
13. Le, D: Higher integrability for gradients of solutions to degenerate parabolic systems. Discr Contin Dyn Syst. 26(2),
597–608 (2010)
14. Ladyzenskaja, OA, Solonnikov, VA, Ural’ceva, NN: Linear and quasilinear equations of parabolic type. Translation of
Mathematical Monographs. American Mathematical Society, Providence. 23 (1968)
15. Friedman, A: Partial Differential Equations of Parabolic Type. Prentice-Hall Inc, Engle-wood Cliffs (1964)
16. Coddington, E, Levinson, N: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955)
17. Anderson, JR: Local existence and uniqueness of solutions of degenerate parabolic equations. Commun Partial Differ
Equ. 16, 105–143 (1991). doi:10.1080/03605309108820753
doi:10.1186/1687-1847-2011-12
Cite this article as: Zhang et al.: Local existence and uniqueness of solutions of a degenerate parabolic system.
Advances in Difference Equations 2011 2011:12.
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