Local Solutions to a Class of Parabolic System Related to the P-Laplacian

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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.

References

[1] Astrita, G. and Marrucci, G. (1974) Principles of Non-Newtonian Fluid Mechanics. McGraw-Hill, New York.

[2] Martinson, L.K. and Pavlov, K.B. (1971) Unsteady Shear Flows of a Conducting Fluid with a Rheological Power Law. Magnitnaya Gidrodinamika, 2, 50-58.

[3] Esteban, J.R. and Vazquez, J.L. (1982) On the Equation of Turbulent Filteration in One-Dimensional Porous Media. Nonlinear Analysis, 10, 1303-1325.

http://dx.doi.org/10.1016/0362-546X(86)90068-4

[4] Constantin, A., Escher, J. and Yin, Z. (2004) Global Solutions for Quasilinear Parabolic System. Journal of Differential Equations, 197, 73-84.

http://dx.doi.org/10.1016/S0022-0396(03)00165-7

[5] Dichstein, F. and Escobedo, M. (2001) A Maximum Principle for Semilinear Parabolic Systems and Application. Nonlinear Analysis, 45, 825-837.

http://dx.doi.org/10.1016/S0362-546X(99)00419-8

[6] Pierre, M. and Schmidt, D. (1997) Blowup in Reaction-Diffusion Systems with Dissipation of Mass. SIAM Journal on Mathematical Analysis, 28, 259-269.

http://dx.doi.org/10.1137/S0036141095295437

[7] Zhao, J. (1993) Existence and Nonexistence of Solutions for . Journal of Mathematical Analysis and Applications, 172, 130-146.

http://dx.doi.org/10.1006/jmaa.1993.1012

[8] Wei, Y. and Gao, W. (2007) Existence and Uniqueness of Local Solutions to a Class of Quasilinear Degenerate Parabolic Systems. Applied Mathematics and Computation, 190, 1250-1257.

http://dx.doi.org/10.1016/j.amc.2007.02.007

[9] Ladyzenskaja, O.A., Solonnikov, V.A. and Uralâ€™ceva, N.N. (1968) Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence, RI.

[10] Friedman, A. (1964) Partial Differential Equations of Parabilic Type. Prentice-Hall Inc., Englewood Cliffs, NJ.

[11] Coddingtin, E. and Levinson, N. (1955) Theory of Ordinary Differential Equations. McGraw-Hill, New York.