Vol. 5, No. 1, 2017, pp 7-11. doi: 10.12691/ajma-5-1-2 | Research Article

Abstract | |

1. | Introduction |

2. | Prerequisites |

3. | Formulation of the Problem |

4. | Existence Theorem |

References |

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

- 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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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

Let be a bounded domain in with a smooth (uniform C^{m}-) 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

A_{1}) is continuous in for almost all and and a fixed function

for all ,all and all .

A_{2}) For all and two distinct

A_{3}) 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

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 *e*stimate

Therefore

Including the existence of a solution

**Theorem.** Let the hypotheses A_{1})-A_{3}) 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 A_{2}) 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 A_{1}) and G), taking Young’s inequality into account, we get

and (4) follows. Assertion (5) is a direct consequence of A_{2}), 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

[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. | ||

In article | |||

[3] | El-Dessouky, A.T., Strongly nonlinear parabolic variational inequalities. J. of Mathematical Analysis and Applications, 181, 1994, 498-504. | ||

In article | View Article | ||

[4] | Landes, R., and Mustonen, V., A strongly nonlinear parabolic initial boundary value problem. Arkiv for Matematik, 25, 1987, 29-40. | ||

In article | View Article | ||

[5] | Landes, R., A note on strongly nonlinear parabolic equations of Higher order. Diff. and Integral equations,Vol.3, No.5, 1990, 851-862. | ||

In article | |||