The objective of this paper is to study the existence and uniqueness of local solutions to the initial and boundary value problem of the parabolic system
where is a bounded domain with smooth boundary. The conditions of and will be given later.
System (1.1) is popular applied in non-Newtonian fluids   and nonlinear filtration  , etc. In the non-Newtonian fluids theory, are all characteristic quantity of the medium. Media with are called dilatant fluids and those with are called pseudoplastics. If, they are Newtonian fluids.
Some authors have studied the global finiteness of the solutions (see   ) and blow-up properties of the solutions (see  ) with various boundary conditions to the systems of evolutionary Laplacian equations. Zhao  and Wei-Gao  studied the existence and blow-up property of the solutions to a single equation and the systems of two equations. We found that the method of  can be extended to the general systems of n equations. For the sake of simplicity, this paper only makes a detailed discussion on n = 3. Since the system is coupled with nonlinear terms, it is in general difficult to study the system. In this paper, we consider some special cases by stating some methods of regularization to construct a sequence of approximation solutions with the help of monotone iteration technique and obtain the existence of solutions to a regularized system of equations. Then we obtain the existence of solutions to the system (1.1)-(1.3) by a standard limiting process. Systems (1.1) degenerates when or. In general, there would be no classical solutions and hence we have to study the generalized solutions to the problem (1.1)-(1.3).
The definition of generalized solutions in this work is the following.
Definition 1.1. Function is called a generalized solution of the system (1.1)-(1.3) if, and satisfies
for any for
Equations (4) implies that
The followings are the constrains to the nonlinear functions involved in this paper.
Definition 1.2. A function is said to be quasimonotone nondecreasing (resp., nonincreasing) if for fixed, is nondecreasing (resp., non- increasing) in
Our main existence result is following:
Theorem 1.3. If there exist nonnegative functions which are quasimonotonically nondecreasing for, , , and a non- negative function such that
Then there exists a constant such that the system (1.1)-(1.3) has a solution in the sence of Definition 1.1 with replaced by.
In Theorem 1.3, we just obtain the existence of local solution. As known to all, when the system degenerates into an equation, as long as some order of growth conditions is added on, we can find the global solution, which is the main result of  . The existence of the global solution of (1.1)-(1.3) remains to be further studied.
On the other hand, similar to  , we made the assumption of monotonicity to. From the current point of view, the condition is relatively strong. It is well worth studying how to reduce monotonicity requirements of the system (1.1)-(1.3).
2. Proof of Theorem 1.3
To prove the theorem, we consider the following regularized problem
where, are quasimonotone nondecreasing and uniformly on bounded subsets of also
Lemma 2.1. The regularized problem (2.1)-(2.3) has a generalized solution.
Proof. Starting from a suitable initial iteration, we construct a se- quence from the iteration process
where. It is clear that for each the above system consists of three nondegenerated and uncoupled initial boundary-value problems.
By classical results (see  ) for fixed and the problem (2.5)-(2.7) has a classical solution if is smooth.
To ensure that this sequence converges to a solution of (2.1)-(2.3), it is necessary to choose a suitable initial iteration. The choice of this function depends on the type of quasimonotone property of. In the following, we establish the monotone property of the sequence.
Set. Let be a classical solution of the following problem.
By and the comparison theorem (see  ), we have that
Hence by the quasimonotone nondecreasing property of, we have
Using the same argument as above, we can obtain a classical solution of the problem
By the comparison theorem, we have
By induction method, we obtain a nonincreasing sequence of smooth functions
In a similar way, by setting we can get a solution of
In the same way as above, we obtain a nondecreasing sequence of smooth functions
It is obvious that. By induction method, we may assume that. Since is quasimonotone nondecreasing, we have
By the comparison principle, we have. Therefore
Taking, we get a nondecreasing bounded sequence
. Hence there exist functions such that
By the continuity of we have
We now prove that there exist and a constant M (independent of k and) such that for all k, we have
Let be the solutions of the ordinary differential equations
By standard results in  , there exist, such that exists on with depends only on. By the comparison theorem
Setting, we obtain (2.31).
We now claim that as, in, where stands for weak convergence,.
Multiplying (2.5) by and integrating over, we obtain that
By (2.12) and the property of
where C is a constant independent of and k.
Multiplying (2.5) by and integrating over, we have
By Cauchy inequality and integrating by parts, we obtain
By (2.37) and (2.40), we obtain that there exists a subsequence of converging weakly in the following sense as.
where stands for weak convergence,.
From (2.29), (2.30), (2.37), (2.40) and the uniqueness of the weak limits, we have that, as,
We now claim that
Multiplying (2.5) by and integrating over with we get
Since the three terms on the right hand side of the above equality converge to 0 as. This yields that
On the other hand, since, we have that
Following (2.50) and (2.51), we have
by Hölder inequality, we have
This proves that any weak convergence subsequence of will have as its weak limit and hence by a standard argument, we have that as,
Combining the above results, we have proved that is a generalized solution of (2.1)-(2.3).
Proof of theorem 1.3.
Since satisfy similar estimates as (2.31), (2.37) and (2.40), combining the property of, we know that there are functions (as) such that for some subsequence of denoted again by,
In a similar way as above, we prove that
By a standard limiting process, we obtain that satisfies the initial and boundary value conditions and the integrating expression. Thus is a generalized solution of (1.1)-(1.3).
3. Uniqueness Result to the Solution of the System
We now prove the uniqueness result to the solution of the system.
Theorem 3.1. Assume is Lipschitz continuous in, then the solution of (1.1)-(1.3) is unique.
Proof. Assume that and are two solutions of (1.1)- (1.3). Let then following (1.5),
By (3.1) subtracting (3.2), we get
By the inequality (3.3) and the Lipschitz condition, a simple calculation shows that
Setting, then (3.4) can be written as. Since, by a standard argument, we have, and hence.