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Anne Lanham, Lillian Mwanri

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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

Keywords: strongly nonlinear parabolic operators-Systems of variational inequalities

Citation: A.T. El-Dessouky, On Weak Solutions of Systems of Strongly Nonlinear Parabolic Variational Inequalities. American Journal of Mathematical Analysis, 5 (1). 7-11. January 2017. doi:10.12691/ajma-5-1-2

1. Introduction

Consider the parabolic initial-boundary value problem

[picture] | (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

[picture] | (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

[picture] | (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

[picture] | (4)

[picture] | (5)

[picture] | (6)

[picture] | (7)

and

[picture] | (8)

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

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

[picture] | (9)

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

Allowing we get

[picture] | (10)

where is the Gateaux derivative of at .

Similarly, from (2), we get

Letting we have

[picture] | (11)

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

Therefore

[picture] | (12)

On the other hand, we get from(2)

In particular,

[picture] | (13)

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

[picture] | (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

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