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# Local existence and uniqueness of solutions of a degenerate parabolic system

- Dazhi Zhang
^{1}, - Jiebao Sun
^{1}Email author and - Boying Wu
^{1}

**2011**:12

https://doi.org/10.1186/1687-1847-2011-12

© Zhang et al; licensee Springer. 2011

**Received:**30 November 2010**Accepted:**16 June 2011**Published:**16 June 2011

## 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

where *m*
_{
i
} > 1*, i* = 1, 2, *Q*
_{
T
} = Ω × (0*, T* ), Ω is a bounded domain in ℝ ^{
N
} with smooth boundary,
and
.

The coupled equations in (1.1) provide a class of quasilinear degenerate parabolic systems. Problems of this form arise in a number of areas of science. For instance, in models for gas or fluid flow in porous media [1–3] and for the spread of certain biological populations [4–6]. When *m*
_{1} = *m*
_{2} = 1, the system (1.1) models the Newtonian fluids, which is couples with Laplace equations. For various initial boundary problems to this kind system, many articles have been devoted to the existence of the solutions and blowup properties of the solutions [7–9].

In recent years, degenerate parabolic systems are of particular interests since they can take into account nonlinear diffusion occurring in the phenomena appearing in the models, and have been extensively studied by many researchers (see e.g., [3, 10–13] and the references therein). The degeneracy and coupled with nonlinear terms of this systems cause great difficulties to study them. In this article, we will establish the local existence and uniqueness results under some special cases for the nonlinear reaction terms. First, by making use the method of regularization and monotone iteration technique, we obtain a sequence of approximation solutions. Then a weak solution is obtained as the limit of the solutions of such problems. Executing this program one encounters two difficulties. The first is proving that the approximating problems which are nondegenerate admits a solution, the second difficulty is to establish 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}) is called to be a weak solution of the problem (1.1)-(1.3) provided that , , and

*φ*

_{ i }|

_{∂Ω×(0, T)}= 0,

*φ*

_{ i }(

*x*,

*T*) = 0,

*i*= 1, 2. The above equation also implies

**Definition 1.2**. A function *f* = *f*(*u*
_{1}
*, u*
_{2}) is said to be quasimonotone nondecreasing (respectively, nonincreasing) if for fixed *u*
_{1} (or *u*
_{2}), *f* is nondecreasing (respectively, nonincreasing) in *u*
_{2} (or *u*
_{1}).

Throughout this article, we assume *f*
_{
i
} (*x, t, u*
_{1}
*, u*
_{2})(*i* = 1, 2) satisfies the following condition:

(A0) *f*
_{
i
} (*x, t, u*
_{1}
*, u*
_{2})(*i* = 1, 2) is quasimonotonically nondecreasing for *u*
_{1}
*, u*
_{2}.

## 2 Existence and uniqueness

In this section, we show the local existence and uniqueness of weak solutions of (1.1)-(1.3). First, we show the local existence results.

**Theorem 2.1**. *Assume (A0), (A1) hold, then there exists a constant T*
_{1} ∈ [0, *T*] *such that (1.1)-(1.3) admits a solution* (*u*
_{1}
*, u*
_{2}) *in*
.

*Proof*. Due to the degeneracy of the system (1.1), we consider the following regularized problem

where
; *f*
_{
iε
} → *f*
_{
i
} uniformly on bounded subsets of
, and *f*
_{
iε
} satisfies the assumptions (A0), (A1),
,
,
, strongly in
as *ε* → 0.

with a suitable initial value
, *i* = 1, 2. By classical results in [14], the problem (2.4)-(2.6) admits a classical solution
for fixed *k* and *ε* when
is smooth. The choice of the initial iteration value which will be obtained by the quasimonotone property of (*f*
_{1}
*, f*
_{2}) would be crucial to ensure that the above sequence converges to a solution of the generalized problem.

*k*= 2, and . So we can obtain a nondecreasing sequence

*f*

_{ iε }is quasimonotone nondecreasing, we have

and the comparison theorem, we have . Further we can obtain (2.7).

*u*

_{ iε }(

*i*= 1, 2) such that

*T*

_{1}∈ (0

*, T*] and a positive constant

*M*(independent of

*ε*and

*k*), such that for all

*k*,

*i*= 1, 2, such that exists on with depends only on . By the comparison theorem, we have

Then by setting and , we obtain (2.10).

Now we show that
in
,
in
as *k* → ∞, where ⇀ stands for weak convergence.

where C is a constant independent of *k, ε*.

*k → ∞*,

*i*= 1, 2. Furthermore (2.11) implies that there exists ,

*s*= 1, ...,

*n*, such that

where *ν* = (*ν*
_{1}, ..., *ν*
_{
n
} ),
with
, *φ*
_{
i
} (*x*, *T*
_{1}) = 0, *i* = 1, 2.

*ζ*(

*x, T*

_{1}) = 0, multiplying (2.4) by and integrating over , we have

where
with
. Obviously, if we let *δ* ≤ 0, we can get the inverted inequality. So we can obtain (2.16) by choosing suitable *ζ*, s.t. supp*φ*
_{
i
} ⊂ supp*ζ* and *ζ* = 1 on supp*φ*
_{
i
} .

In summary, we have proved that *u*
_{
ε
} = (*u*
_{1ε
}, *u*
_{2ε
}) is a weak solution of (2.1)-(2.3).

*u*

_{ ε }= (

*u*

_{1ε },

*u*

_{2ε }) is a weak solution of (1.1)-(1.3). Since

*u*

_{ ε }= (

*u*

_{1ε },

*u*

_{2ε }) satisfies similar estimates as (2.10)-(2.12), combining the property of

*f*

_{ iε }, we know that there are functions , ,

*i*= 1, 2, such that for some subsequence of (

*u*

_{1ε },

*u*

_{2ε }), denoted by itself for simplicity, when

*ε*→ 0

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}) *is Lipschitz continuous in* (*u*
_{1}
*, u*
_{2})*, then* (1.1)-(1.3) *has a unique solution*.

*Proof*. Assume that

*u*= (

*u*

_{1}

*, u*

_{2}),

*v*= (

*v*

_{1}

*, v*

_{2}) are two solutions of (1.1)-(1.3). Form Definition 1, we see that

*u*

_{1}

*, u*

_{2}) and (

*v*

_{1}

*, v*

_{2}) are bounded on

*Q*

_{ t }, it follows from

*m >*1, Φ(

*x, s*) is a bounded nonnegative function. Thus, appropriate test function

*φ*

_{ i }may be chosen exactly as in [[17], pp. 118-123] and combined with the Lipschitz continuity of

*f*

_{ i }to obtain

Combined with the Gronwall's lemma, we see that *u*
_{
i
} ≡ *v*
_{
i
} , *i* = 1, 2. The proof is completed. □

## Declarations

### Acknowledgements

The authors express their deep thanks to the referees for their very helpful suggestions to improve some results in this paper. This work is supported by “the Fundamental Research Funds for the Central Universities” (Grant No. HIT. NSRIF. 2011006) and also by the 985 project of Harbin Institute of Technology.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.