Journal of Applied Mathematics

Volume 2012 (2012), Article ID 936140, 17 pages

doi:10.1155/2012/936140

Research Article

Asymptotic Behavior for a Nondissipative and Nonlinear System of the Kirchhoff Viscoelastic Type

Nasser-Eddine Tatar

Department of Mathematics & Statistics, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia

Received 3 June 2012; Accepted 17 July 2012

Academic Editor: Yongkun Li

Copyright © 2012 Nasser-Eddine Tatar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

A wave equation of the Kirchhoff type with several nonlinearities is stabilized by a viscoelastic damping. We consider the case of nonconstant (and unbounded) coefficients. This is a nondissipative case, and as a consequence the nonlinear terms cannot be estimated in the usual manner by the initial energy. We suggest a way to get around this difficulty. It is proved that if the solution enters a certain region, which we determine, then it will be attracted exponentially by the equilibrium.

1. Introduction

We will consider the following wave equation with a viscoelastic damping term:

where is a bounded domain in with smooth boundary and . The functions and are given initial data, and the (nonnegative) functions , and are at least absolutely continuous and will be specified later on. This problem arises in viscoelasticity where it has been shown by experiments that when subject to sudden changes, the viscoelastic response not only does depend on the current state of stress but also on all past states of stress. This gives rise to the integral term called the memory term. One may find a rich literature in this regard (with or without the Kirchhoff terms) treating mainly the stabilization of such systems for different classes of functions . We refer the reader to [1–25] and the references therein. For problems of the Kirchhoff type, one can consult [26–35] and in particular [36–46] where the equations are supplemented by a nonlinear source. Several questions, such as well-posedness and asymptotic behavior, have been discussed in these references, to cite but a few.

As is clear from the equation in (1.1), we consider here several nonlinearities and the relaxation function is not necessarily decreasing or even nonincreasing. These issues are important but do not constitute the main contribution in the present paper. In case that and are not nonincreasing, then we are in a nondissipative situation. This is the case also when the relaxation function oscillates (in case are nonincreasing). Our argument here is simple and flexible. It relies on a Gronwall-type inequality involving several nonlinearities. We prove that there exists a sufficiently large and a constant after which (the modified energy of) global solutions are bounded below by or decay to zero exponentially. We were not able to find conditions directly on the initial data because the Gronwall inequality is applicable only after some large values of time.

For simplicity we shall consider the simpler case and .

The local existence and uniqueness may be found in [36, 37].

Theorem 1.1. Assume that and is a nonnegative summable kernel. If when and when , then there exists a unique solution to problem (1.1) such that

for small enough.

The plan of the paper is as follows. In the next section we prepare some materials needed to prove our result. Section 3 is devoted to the statement and proof of our theorem.

2. Preliminaries

In this section we define the different functionals we will work with. We prove an equivalence result between two functionals. Further, some useful lemmas are presented. We define the (classical) energy by

where denotes the norm in . Then by  (1.1)  it is easy to see that for

The first term in the right-hand side of (2.2) may be written as the derivative of some expression; namely,

where

Therefore, if we modify to

we obtain for

Assuming that

makes a nonnegative functional. The following functionals

are standard and will be used here. The next ones have been introduced by the present author in [24]

where

and and are two nonnegative functions which will be precised later (see (H2), (H3)). The functional

for some , to be determined is equivalent to .

Proposition 2.1. There exist such that

for all and small .

Proof. By the inequalities

where is the Poincaré constant, we have

On the other hand,

Therefore, for some constant and small such that and .

The identity to follow is easy to justify and is helpful to prove our result.

Lemma 2.2. One has for and

The next lemma is crucial in estimating (partially) our nonlinear terms. It can be found in [47].

Let , and let . We write if is nondecreasing in .

Lemma 2.3. Let be a positive continuous function in are nonnegative continuous functions, are nondecreasing continuous functions in , with for , and is a nonnegative continuous functions in . If in , then the inequality

implies that

where ,

and is chosen so that the functions are defined for .

Lemma 2.4. Assume that if or if . Then there exists a positive constant such that

for .

3. Asymptotic Behavior

In this section we state and prove our result. To this end we need some notation. For every measurable set , we define the probability measure by

The nondecreasingness set and the non-decreasingness rate of are defined by

respectively.

The following assumptions on the kernel will be adopted.

for all and . is absolutely continuous and of bounded variation on and for some nonnegative summable function (= where exists) and almost all . There exists a nondecreasing function such that is a nonincreasing function: and .

Note that a wide class of functions satisfies the assumption (H3). In particular, exponentially and polynomially (or power type) decaying functions are in this class.

Let be a number such that . We denote by the set .

Lemma 3.1. One has for and

where is the total variation of .

Proof. This lemma is proved by a direct differentiation of along solutions of (1.1) and estimation of the different terms in the obtained expression of the derivative. Indeed, we have

or

Therefore,

For all measurable sets and such that , it is clear that

For , the first term in the right-hand side of (3.8) satisfies

and the third one fulfills

Back to (3.8) we may write

The last term in the right-hand side of (3.7) will be estimated as follows:

For the fourth term in (3.7), it holds that

Moreover, from Lemma 2.4, for if and if , we find

The definition of in (2.5) allows us to write

Gathering all the relations (3.11)–(3.15) together with (3.7), we obtain for

In the following theorem we will assume that just to fix ideas. The result is also valid for . It suffices to interchange and in the proof following it. The case is easier.

We will make use of the following hypotheses for some positive constants , and to be determined.

is a continuously differentiable function such that . is a continuously differentiable function such that . if and if .. .

Theorem 3.2. Assume that the hypotheses (H1)–(H3), (A)–(C) hold and . If , then, for global solutions and small , there exist and such that or

for some positive constants and as long as (D) holds. If , then there exist and such that or

for some positive constants and as long as (E) holds.

Proof. A differentiation of with respect to along trajectories of (1.1) gives

and Lemma 2.2 implies

Next, a differentiation of and yields

Taking into account Lemma 3.1 and the relations (2.6), (3.20)-(3.21), we see that

Next, as in [17], we introduce the sets

and observe that

where is the null set where is not defined and is as in (3.2). Furthermore, if we denote , then because for all and . Moreover, we designate by the sets

In (3.22), we take and . Choosing , it is clear that

for small and , large and , if . We deduce that

Furthermore, if , then

with

and a small . Pick

and such that

Note that this is possible if is so large that even though

Taking the relations (3.22)–(3.30) into account and selecting so that

and small enough so that

we find for , large , small , and

for some positive constant . Take small, and

(i.e., for some ) and

(i.e., for some ) to derive that

for some positive constant .

If , then there exist a and such that for . Thus, in virtue of Proposition 2.1, for , we have

where

If there exists a such that , where , then from (3.39)

and it follows that for

with . Now we apply Lemma 2.3 to get

where and . If, in addition, , then is uniformly bounded by a positive constant . Thus

and by continuity (3.44) holds for all .

If , then for any there exists a such that for . Therefore,

for some . The previous argument carries out with replaced by .

In case that , we reverse the roles of and in the argument above. The case is clear.

Remark 3.3. The case where the derivative of the kernel does not approach zero on (as is the case, for instance, when on is interesting. Indeed, the right-hand side in condition (3.34) will be replaced by with a possibly large constant .

Remark 3.4. The argument clearly works for all kinds of kernels previously treated where derivatives cannot be positive or even take the value zero. In these cases there will be no need for the smallness conditions on the kernels. This work shows that derivatives may be positive (i.e., kernels may be increasing) on some “small” subintervals and open the door for (optimal) estimations and improvements of these sets.

Remark 3.5. The assumptions and may be relaxed to and , respectively. In this case and would depend on .

Remark 3.6. The assertion in Theorem 3.2 is an “alternative” statement. As a next step it would be nice to discuss the (sufficient conditions of) occurrence of each case in addition to the global existence.

Acknowledgment

The author is grateful for the financial support and the facilities provided by the King Fahd University of Petroleum and Minerals.

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