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In this paper, the existence and uniqueness of local solutions to the initial and boundary value problem of a class of parabolic system related to the p-Laplacian are studied. The regularization method is used to construct a sequence of approximation solutions, with the help of monotone iteration technique, then we get the existence of solution of a regularized system. By the use of a standard limiting process, the existence of the local solutions of the system is obtained. Finally, the uniqueness of the solution is also proven.

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

System (1.1) is popular applied in non-Newtonian fluids [

Some authors have studied the global finiteness of the solutions (see [

The definition of generalized solutions in this work is the following.

Definition 1.1. Function

for any

Equations (4) implies that

The followings are the constrains to the nonlinear functions

Definition 1.2. A function

Our main existence result is following:

Theorem 1.3. If there exist nonnegative functions

Then there exists a constant

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

On the other hand, similar to [

To prove the theorem, we consider the following regularized problem

where

Lemma 2.1. The regularized problem (2.1)-(2.3) has a generalized solution.

Proof. Starting from a suitable initial iteration

where

By classical results (see [

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

Set

By

Hence by the quasimonotone nondecreasing property of

for

Using the same argument as above, we can obtain a classical solution

for

By the comparison theorem, we have

By induction method, we obtain a nonincreasing sequence of smooth functions

In a similar way, by setting

with

In the same way as above, we obtain a nondecreasing sequence of smooth functions

It is obvious that

for

By the comparison principle, we have

Taking

By the continuity of

We now prove that there exist

Let

By standard results in [

Setting

We now claim that

Multiplying (2.5) by

Furthermore

By (2.12) and the property of

where C is a constant independent of

Multiplying (2.5) by

By Cauchy inequality and integrating by parts, we obtain

Hence

By (2.37) and (2.40), we obtain that there exists a subsequence of

where

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

Hence

Since the three terms on the right hand side of the above equality converge to 0 as

On the other hand, since

Note that

Following (2.50) and (2.51), we have

Since

and

by Hölder inequality, we have

i.e.,

Hence

This proves that any weak convergence subsequence of

Combining the above results, we have proved that

Proof of theorem 1.3.

Since

In a similar way as above, we prove that

By a standard limiting process, we obtain that

We now prove the uniqueness result to the solution of the system.

Theorem 3.1. Assume

Proof. Assume that

By (3.1) subtracting (3.2), we get

By the inequality (3.3) and the Lipschitz condition, a simple calculation shows that

Setting

Ou, Q.T. and Zhan, H.S. (2016) Local Solutions to a Class of Parabolic System Related to the P-Lapla- cian. Advances in Pure Mathematics, 6, 868- 877. http://dx.doi.org/10.4236/apm.2016.612065