Local existence and uniqueness of solutions of a degenerate parabolic system

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.

1 Introduction

In this article, we consider the following degenerate parabolic system

(1.1)

(1.2)

(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, 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

for any test function with 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.

(A1) There exists a nonnegative function g(u) ∈ C ¹(ℝ) such that

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

(2.1)

(2.2)

(2.3)

where ; f _iε → f _i uniformly on bounded subsets of , and f _iε satisfies the assumptions (A0), (A1), , , , strongly in as ε → 0.

Now we will prove that the regularized problem (2.1)-(2.3) admits a classical solution. Construct a sequence from the following iteration process

(2.4)

(2.5)

(2.6)

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.

Let , and be a classical solution of the following problem

By the comparison theorem [15], we have

Then the quasimonotone nondecreasing property of f _iε shows that

Then we can also obtain a classical solution from (2.4)-(2.6) when k = 2, and . So we can obtain a nondecreasing sequence

With the similar method, by setting , we obtain a classical solution of the following problem

and

And the quasimonotone nondecreasing property of f _iε also shows that

Now we show

(2.7)

It is obvious that . Assume that , we just need to prove that . Since f _iε is quasimonotone nondecreasing, we have

From the iteration equations

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

Let , then is a nondecreasing bounded sequence. Then there exist functions u _iε (i = 1, 2) such that

(2.8)

The continuity of function f _iε (i = 1, 2) also shows that

(2.9)

Therefore, we claim that there exist T ₁ ∈ (0, T] and a positive constant M (independent of ε and k), such that for all k,

(2.10)

Let be the solutions of the ordinary differential equations

The results in [16] show that there exists , 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.

Multiplying (2.4) by and integrating over , we have

that is

Then by (2.10) and the property of f _iε , we have

(2.11)

where C is a constant independent of k, ε.

Multiplying (2.4) by and integrating over , by Young's inequality we have

Noticing that the first term of the left side of the above inequality can be rewritten as

Then we have

Therefore

Furthermore, we can obtain

(2.12)

Following (2.8), (2.9), (2.12) and the uniqueness of the weak limits, it is easy to know that, as k → ∞,

(2.13)

(2.14)

where ⇀ stands for weak convergence, i = 1, 2. Furthermore (2.11) implies that there exists , s = 1, ..., n, such that

Hence,

(2.15)

where ν = (ν ₁, ..., ν _n ), with , φ _i (x, T ₁) = 0, i = 1, 2.

Now for any φ _i given as before, we show

(2.16)

For any , , 0 ≤ ζ ≤ 1, with ζ(x, T ₁) = 0, multiplying (2.4) by and integrating over , we have

(2.17)

Notice that

from (2.17), we get

Letting k → ∞, then

(2.18)

Set in (2.15), we obtain

Substituting the above equation into (2.18), we get

(2.19)

Taking , δ ≥ 0 in (2.19) and then let δ → 0, we obtain

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. suppφ _i ⊂ suppζ and ζ = 1 on suppφ _i .

In summary, we have proved that u _ε = (u _1ε , u _2ε ) is a weak solution of (2.1)-(2.3).

Now, we will prove that the limit of 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

(2.20)

(2.21)

Subtracting the two equations, we get

(2.22)

where

Since (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

where C > 0 is a bounded constant. Further, we have

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