Skip to content

Advertisement

  • Research
  • Open Access

Local existence and uniqueness of solutions of a degenerate parabolic system

Advances in Difference Equations20112011:12

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

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

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
(1.1)
(1.2)
(1.3)

where m i > 1, i = 1, 2, Q T = Ω × (0, T ), Ω is a bounded domain in 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 [13] and for the spread of certain biological populations [46]. 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 [79].

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, 1013] 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() 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 f i uniformly on bounded subsets of , and f 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 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 also shows that
Now we show
(2.7)
It is obvious that . Assume that , we just need to prove that . Since f 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 (i = 1, 2) such that
(2.8)
The continuity of function f (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 , 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 , 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.    □

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

(1)
Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, PR China

References

  1. Arossox DG, Craxdall MG, Peletier LA: Stabilization of solutions of a degenerate nonlinear diffusion problem. Nonlinear Anal 1982,6(10):1001-1022. 10.1016/0362-546X(82)90072-4MathSciNetView ArticleGoogle Scholar
  2. Bear J: Dynamics of Fluids in Porous Media. Elsevier, New York; 1972.Google Scholar
  3. Lei PD, Zheng SN: Global and nonglobal weak solutions to a degenerate parabolic system. J Math Anal Appl 2006,324(1):177-198. 10.1016/j.jmaa.2005.12.012MathSciNetView ArticleMATHGoogle Scholar
  4. Okubo A: Diffusion and Ecological Problems: Mathematical Models. In Biomathematics. Volume 10. Springer, Berlin, Heidelberg, New York; 1980.Google Scholar
  5. Meinhardt H: Models of Biological Pattern Formation. Academic Press, London; 1982.Google Scholar
  6. Romanvoskii YM, Stepanova NV, Chernavskii DS: Mathematical biophysics. Nauka, Moscow 1984. (in Russian)Google Scholar
  7. Constantin A, Escher J, Yin Z: Global solutions for quasilinear parabolic system. J Differ Equ 2004,197(1):73-84. 10.1016/S0022-0396(03)00165-7MathSciNetView ArticleMATHGoogle Scholar
  8. Dickstein F, Escobedo M: A maximum principle for semilinear parabolic systems and application. Nonlinear Anal 2001,45(7):825-837. 10.1016/S0362-546X(99)00419-8MathSciNetView ArticleMATHGoogle Scholar
  9. Pierre M, Schmidt D: Blowup in reaction diffusion systems with dissipation of mass. SIAM J Math Anal 1997,28(2):259-269. 10.1137/S0036141095295437MathSciNetView ArticleMATHGoogle Scholar
  10. Wang MX, Wei YF: Blow-up properties for a degenerate parabolic system with nonlinear localized sources. J Math Anal Appl 2008,343(2):621-635. 10.1016/j.jmaa.2008.01.073MathSciNetView ArticleMATHGoogle Scholar
  11. Cui ZJ, Yang ZD: Boundedness of global solutions for a nonlinear degenerate parabolic (porous medium) system with localized sources. Appl Math Comput 2008,198(2):882-895. 10.1016/j.amc.2007.09.037MathSciNetView ArticleMATHGoogle Scholar
  12. Litcanu G, Morales-Rodrigo C: Global solutions and asymptotic behavior for a parabolic degenerate coupled system arising from biology. Nonlinear Anal 2008,72(1):77-98.MathSciNetView ArticleMATHGoogle Scholar
  13. Le D: Higher integrability for gradients of solutions to degenerate parabolic systems. Discr Contin Dyn Syst 2010,26(2):597-608.View ArticleMathSciNetMATHGoogle Scholar
  14. Ladyzenskaja OA, Solonnikov VA, Ural'ceva NN: Linear and quasilinear equations of parabolic type. Translation of Mathematical Monographs. American Mathematical Society, Providence 1968., 23: Google Scholar
  15. Friedman A: Partial Differential Equations of Parabolic Type. Prentice-Hall Inc, Engle-wood Cliffs; 1964.Google Scholar
  16. Coddington E, Levinson N: Theory of Ordinary Differential Equations. McGraw-Hill, New York; 1955.Google Scholar
  17. Anderson JR: Local existence and uniqueness of solutions of degenerate parabolic equations. Commun Partial Differ Equ 1991, 16: 105-143. 10.1080/03605309108820753View ArticleMathSciNetMATHGoogle Scholar

Copyright

© Zhang et al; licensee Springer. 2011

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.

Advertisement