APMAdvances in Pure Mathematics2160-0368Scientific Research Publishing10.4236/apm.2016.612065APM-72060ArticlesPhysics&Mathematics Local Solutions to a Class of Parabolic System Related to the P-Laplacian QitongOu1HuashuiZhan1School of Applied Mathematics, Xiamen University of Technology, Xiamen, China081120160612868877September 27, 2016Accepted: November 14, November 17, 2016© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

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.

Existence Uniqueness Evolution P-Laplacian Parabolic System
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

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

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

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

for.

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

for.

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

with

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

for.

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

Furthermore

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

Hence

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

Hence

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

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 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.

Cite this paper

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

ReferencesAstrita, G. and Marrucci, G. (1974) Principles of Non-Newtonian Fluid Mechanics. McGraw-Hill, New York.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.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)90068-4Constantin, 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)00165-7Dichstein, 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)00419-8Pierre, 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/S0036141095295437Zhao, J. (1993) Existence and Nonexistence of Solutions for . Journal of Mathematical Analysis and Applications, 172, 130-146. http://dx.doi.org/10.1006/jmaa.1993.1012Wei, 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.2007.02.007Ladyzenskaja, O.A., Solonnikov, V.A. and Ural’ceva, N.N. (1968) Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence, RI.Friedman, A. (1964) Partial Differential Equations of Parabilic Type. Prentice-Hall Inc., Englewood Cliffs, NJ.Coddingtin, E. and Levinson, N. (1955) Theory of Ordinary Differential Equations. McGraw-Hill, New York.