On Weak Solutions of Systems of Strongly Nonlinear Parabolic Variational Inequalities

A.T. El-Dessouky

American Journal of Mathematical Analysis

On Weak Solutions of Systems of Strongly Nonlinear Parabolic Variational Inequalities

A.T. El-Dessouky

Mathematics Department, Helwan University, Faculty of Science, Cairo, Egypt

Abstract

In this paper we prove the existence of weak solutions for systems of variational inequalities of strongly nonlinear parabolic operators: where

Cite this article:

  • A.T. El-Dessouky. On Weak Solutions of Systems of Strongly Nonlinear Parabolic Variational Inequalities. American Journal of Mathematical Analysis. Vol. 5, No. 1, 2017, pp 7-11. http://pubs.sciepub.com/ajma/5/1/2
  • El-Dessouky, A.T.. "On Weak Solutions of Systems of Strongly Nonlinear Parabolic Variational Inequalities." American Journal of Mathematical Analysis 5.1 (2017): 7-11.
  • El-Dessouky, A. (2017). On Weak Solutions of Systems of Strongly Nonlinear Parabolic Variational Inequalities. American Journal of Mathematical Analysis, 5(1), 7-11.
  • El-Dessouky, A.T.. "On Weak Solutions of Systems of Strongly Nonlinear Parabolic Variational Inequalities." American Journal of Mathematical Analysis 5, no. 1 (2017): 7-11.

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1. Introduction

Consider the parabolic initial-boundary value problem

(1)

in

where

and . If the coefficients satisfy a polynomial growth conditions of order (p-1) in u and its space derivatives but g obeys no growth in u, but merely a sign condition, the existence of weak solutions problems of the type (1) has been obtained by many authors (cf [1, 4] and [5]). In [2] Browder and Br𝐞́zis extended the above results to the corresponding class of variational inequalities. Their proof based on a type of compactness result. Our result can be viewed as a generalization to systems of variational inequalities for the work of [4] and [5].Our proof relies on deriving a-priori bound for the time derivative of the solution in

2. Prerequisites

Let be a bounded domain in with a smooth (uniform Cm-) boundary, and a positive integer . Denote by

the Sobolev space

where

Let , be its topological dual and with the norm

For the Galerkin method, construct a sequence such that with is dense in Denote by Since is continuously embedded in , which is a Banach space with the norm

then for any there exists a sequence such that in Moreover, since the closure of with respect to the -topology contains , then for any there exists a sequence such that in in the weak sense [5].

We introduce the following hypotheses for

A1) is continuous in for almost all and and a fixed function

for all ,all and all .

A2) For all and two distinct

A3) There exists a constant and a fixed function such that for all and all

G) is continuous in for almost all and measurable in for all Moreover, each is nondecreasing in r for fixed and each for all

3. Formulation of the Problem

Write

By G) each as a function of r is convex, nonnegative and once differentiable.

For set

Let K be a closed convex subset of V containing the origin. Define a proper lower semicontinuous Gateaux differentiable function

Definition: A function is called a Galerkin solution of the associated variational inequalities for (1) if

(2)

where

The existence of a Galerkin solution and its main property, in view of our hypotheses,will be given by the following lemma [3].

Lemma: For every there exists a Galerkin solution such that

Proof: Define a vector-valued function by

in view of our hypotheses, is continuous and coercive an each is lower semicontinuous with

Therefore the system

(3)

for , with

has a local solution.

From (3) we get the estimate

Therefore

Including the existence of a solution

4. Existence Theorem

Theorem. Let the hypotheses A1)-A3) and G) be satisfied. Let be given. Then

(i) there exists with such that

for every for which .

(ii) there exists with such that

for every with

Proof of (i): By the above lemma, there exist Galerkin solutions of (2) such that

Set in (2) we get the uniform boundedness from above of the numerical sequence . The proof will follow if we can show the following assertions for some subsequence of

(4)
(5)
(6)
(7)

and

(8)

To show (4): Given , any and any put

Since is arbitrary, is absolutely for a given Then (2) yields

(9)

In particular, since is arbitrary we can write (9) in the form

Allowing we get

(10)

where is the Gateaux derivative of at .

Similarly, from (2), we get

Letting we have

(11)

Adding (10), (11) and integrating over , taking into account A2) we get

Therefore

(12)

On the other hand, we get from(2)

In particular,

(13)

From (12) and (13),we may apply Gronwall’s inequality to get the estimate

(14)

Using A1) and G), taking Young’s inequality into account, we get

and (4) follows. Assertion (5) is a direct consequence of A2), G) and Aubin’s lemma. Assertion (8) follows from the lower semicontinuity of

To prove (6) and (7), it suffices to show

(15)

Since for any we may find a subsequence such that weakly in we get from (2)

Letting keeping k fixed we have

Since the left hand side of this inequality is independent of , allowing we get (15) and (i) of the theorem follows.

To prove (ii) little arguments are needed. For this aim, define the truncated perturbation (x,t;u)by

From (i), there exists with such that

for every for which

where

and

Using the subgradient inequality for as a function of we have

The rest of the proof is more or less word for word as in (i).

Example: As an example which can be handled by our result, consider the variational inequalities associated with the following system

References

[1]  Br𝐞́zis, H. and Browder, F.E., Strongly nonlinear parabolic initial boundary value problems.Proc.Nat.Acad.Sci.U.S.A.76, 1979, 38-40.
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[2]  Browder, F.E., and Br𝐞́zis, H., Strongly nonlinear parabolic variational inequalities. Proc. Nat. Acad. Sci. U.S.A .77, 2, 1980, 713-715.
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[3]  El-Dessouky, A.T., Strongly nonlinear parabolic variational inequalities. J. of Mathematical Analysis and Applications, 181, 1994, 498-504.
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[4]  Landes, R., and Mustonen, V., A strongly nonlinear parabolic initial boundary value problem. Arkiv for Matematik, 25, 1987, 29-40.
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[5]  Landes, R., A note on strongly nonlinear parabolic equations of Higher order. Diff. and Integral equations,Vol.3, No.5, 1990, 851-862.
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