Local Solutions to a Class of Parabolic System Related to the P-Laplacian ()
1. Introduction
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
(1.1)
(1.2)
(1.3)
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 [1] [2] and nonlinear filtration [3] , 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 [4] [5] ) and blow-up properties of the solutions (see [6] ) with various boundary conditions to the systems of evolutionary Laplacian equations. Zhao [7] and Wei-Gao [8] 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 [8] 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
(1.4)
for any
for ![]()
Equations (4) implies that
(1.5)
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
(1.6)
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 [7] . The existence of the global solution of (1.1)-(1.3) remains to be further studied.
On the other hand, similar to [8] , 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
(2.1)
(2.2)
(2.3)
where
,
are quasimonotone nondecreasing and
uniformly on bounded subsets of
also
(2.4)
strongly in
.
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
(2.5)
(2.6)
(2.7)
where
. It is clear that for each
the above system consists of three nondegenerated and uncoupled initial boundary-value problems.
By classical results (see [9] ) 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.
(2.8)
(2.9)
(2.10)
By
and the comparison theorem (see [10] ), we have that
(2.11)
Hence by the quasimonotone nondecreasing property of
, we have
(2.12)
for
.
Using the same argument as above, we can obtain a classical solution
of the problem
(2.13)
(2.14)
(2.15)
for
.
By the comparison theorem, we have
(2.16)
By induction method, we obtain a nonincreasing sequence of smooth functions
(2.17)
In a similar way, by setting
we can get a solution
of
(2.18)
(2.19)
(2.20)
with
(2.21)
In the same way as above, we obtain a nondecreasing sequence of smooth functions
(2.22)
It is obvious that
. By induction method, we may assume that
. Since
is quasimonotone nondecreasing, we have
(2.23)
for
.
(2.24)
(2.25)
(2.26)
(2.27)
By the comparison principle, we have
. Therefore
(2.28)
Taking
, we get a nondecreasing bounded sequence
. Hence there exist functions
such that
(2.29)
By the continuity of
we have
(2.30)
We now prove that there exist
and a constant M (independent of k and
) such that for all k, we have
(2.31)
Let
be the solutions of the ordinary differential equations
(2.32)
By standard results in [11] , there exist
, such that
exists on
with
depends only on
. By the comparison theorem
(2.33)
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
(2.34)
Furthermore
(2.35)
(2.36)
By (2.12) and the property of ![]()
(2.37)
where C is a constant independent of
and k.
Multiplying (2.5) by
and integrating over
, we have
(2.38)
By Cauchy inequality and integrating by parts, we obtain
(2.39)
Hence
(2.40)
By (2.37) and (2.40), we obtain that there exists a subsequence of
converging weakly in the following sense as
.
(2.41)
(2.42)
(2.43)
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
,
(2.44)
(2.45)
(2.46)
We now claim that ![]()
Multiplying (2.5) by
and integrating over
with
we get
(2.47)
Hence
(2.48)
Since the three terms on the right hand side of the above equality converge to 0 as
. This yields that
(2.49)
On the other hand, since
, we have that
(2.50)
Note that
(2.51)
Following (2.50) and (2.51), we have
(2.52)
Since
(2.53)
and
(2.54)
by Hölder inequality, we have
(2.55)
i.e.,
(2.56)
Hence
(2.57)
This proves that any weak convergence subsequence of
will have
as its weak limit and hence by a standard argument, we have that as
,
(2.58)
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
,
(2.59)
(2.60)
(2.61)
(2.62)
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),
(3.1)
(3.2)
By (3.1) subtracting (3.2), we get
(3.3)
By the inequality (3.3) and the Lipschitz condition, a simple calculation shows that
(3.4)
Setting
, then (3.4) can be written as
. Since
, by a standard argument, we have
, and hence
.