Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2012 (2012), Article ID 936140, 17 pages
http://dx.doi.org/10.1155/2012/936140
Research Article

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

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: 𝑢𝑡𝑡+𝑚𝑖=1𝑏𝑖(𝑡)|𝑢|𝑝𝑖𝑢=1+𝑘𝑗=1𝑎𝑗(𝑡)𝑢2𝑞𝑗2Δ𝑢𝑡0(𝑡𝑠)Δ𝑢(𝑠)𝑑𝑠,inΩ×𝐑+,𝑢=0,onΓ×𝐑+,𝑢(𝑥,0)=𝑢0(𝑥),𝑢𝑡(𝑥,0)=𝑢1(𝑥),inΩ,(1.1) where Ω is a bounded domain in 𝐑𝑛 with smooth boundary Γ=𝜕Ω and 𝑝𝑖,𝑞𝑗>0,𝑖=1,,𝑚,𝑗=1,,𝑘. The functions 𝑢0(𝑥) and 𝑢1(𝑥) are given initial data, and the (nonnegative) functions 𝑎𝑗(𝑡),𝑏𝑖(𝑡), and (𝑡) 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 [125] and the references therein. For problems of the Kirchhoff type, one can consult [2635] and in particular [3646] 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 𝑎𝑗(𝑡) and 𝑏𝑖(𝑡) are not nonincreasing, then we are in a nondissipative situation. This is the case also when the relaxation function oscillates (in case 𝑎𝑗(𝑡),𝑏𝑖(𝑡) 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 𝑇>0 and a constant 𝑈 after which (the modified energy of) global solutions are bounded below by 𝑈 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 𝑝1=𝑝,𝑝𝑖=0,𝑏1=𝑏,𝑏𝑖=0,𝑖=2,,𝑚 and 𝑞1=𝑞,𝑞𝑗=0,𝑎1=𝑎,𝑎𝑗=0,𝑗=2,,𝑘.

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

Theorem 1.1. Assume that (𝑢0,𝑢1)𝐻10(Ω)×𝐿2(Ω) and (𝑡) is a nonnegative summable kernel. If 0<𝑝<2/(𝑛2) when 𝑛3 and 𝑝>0 when 𝑛=1,2, then there exists a unique solution 𝑢 to problem (1.1) such that []𝑢𝐶0,𝑇;𝐻10(Ω)𝐶1[]0,𝑇;𝐿2(Ω)(1.2) for 𝑇 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 1𝐸(𝑡)=2𝑢𝑡22+𝑢22+𝑎(𝑡)2(𝑞+1)𝑢22(𝑞+1)+𝑏(𝑡)𝑝+2𝑢𝑝+2𝑝+2,𝑡0,(2.1) where 𝑝 denotes the norm in 𝐿𝑝(Ω). Then by  (1.1)  it is easy to see that for 𝑡0𝐸(𝑡)=Ω𝑢𝑡𝑡0𝑎(𝑡𝑠)𝑢(𝑠)𝑑𝑠𝑑𝑥+(𝑡)2(𝑞+1)𝑢22(𝑞+1)+𝑏(𝑡)𝑝+2𝑢𝑝+2𝑝+2.(2.2) The first term in the right-hand side of (2.2) may be written as the derivative of some expression; namely, Ω𝑢𝑡𝑡01(𝑡𝑠)𝑢(𝑠)𝑑𝑠𝑑𝑥=2Ω1𝑢𝑑𝑥2(𝑡)𝑢2212𝑑𝑑𝑡Ω(𝑢)𝑑𝑥𝑡0(𝑠)𝑑𝑠𝑢22,(2.3) where (𝑣)(𝑡)=𝑡0||||(𝑡𝑠)𝑣(𝑡)𝑣(𝑠)2𝑑𝑠.(2.4) Therefore, if we modify 𝐸(𝑡) to 1(𝑡)=2𝑢𝑡22+1𝑡0(𝑠)𝑑𝑠𝑢22+Ω(+𝑢)𝑑𝑥𝑎(𝑡)2(𝑞+1)𝑢22(𝑞+1)+𝑏(𝑡)𝑝+2𝑢𝑝+2𝑝+2,(2.5) we obtain for 𝑡01(𝑡)=2Ω||||𝑢(𝑡)𝑢2𝑎𝑑𝑥+(𝑡)2(𝑞+1)𝑢22(𝑞+1)+𝑏(𝑡)𝑝+2𝑢𝑝+2𝑝+2.(2.6) Assuming that 10+(𝑠)𝑑𝑠=1𝜅>0(2.7) makes (𝑡) a nonnegative functional. The following functionals Φ1(𝑡)=Ω𝑢𝑡Φ𝑢𝑑𝑥,2(𝑡)=Ω𝑢𝑡𝑡0(𝑡𝑠)(𝑢(𝑡)𝑢(𝑠))𝑑𝑠𝑑𝑥(2.8) are standard and will be used here. The next ones have been introduced by the present author in [24] Φ3(𝑡)=𝑡0𝐻𝛾(𝑡𝑠)𝑢(𝑠)22𝑑𝑠,Φ4(𝑡)=𝑡0Ψ𝛾(𝑡𝑠)𝑢(𝑠)22𝑑𝑠,(2.9) where 𝐻𝛾(𝑡)=𝛾(𝑡)1𝑡(𝑠)𝛾(𝑠)𝑑𝑠,Ψ𝛾(𝑡)=𝛾(𝑡)1𝑡𝜉(𝑠)𝛾(𝑠)𝑑𝑠,𝑡0,(2.10) and 𝛾(𝑡) and 𝜉(𝑡) are two nonnegative functions which will be precised later (see (H2), (H3)). The functional 𝐿(𝑡)=(𝑡)+4𝑖=1𝜆𝑖Φ𝑖(𝑡)(2.11) for some 𝜆𝑖>0,𝑖=1,2,3,4, to be determined is equivalent to (𝑡)+Φ3(𝑡)+Φ4(𝑡).

Proposition 2.1. There exist 𝜌𝑖>0,𝑖=1,2 such that 𝜌1(𝑡)+Φ3(𝑡)+Φ4(𝑡)𝐿(𝑡)𝜌2(𝑡)+Φ3(𝑡)+Φ4(𝑡)(2.12) for all 𝑡0 and small 𝜆𝑖,𝑖=1,2.

Proof. By the inequalities Φ1(𝑡)=Ω𝑢𝑡1𝑢𝑑𝑥2𝑢𝑡22+𝐶𝑝2𝑢22,Φ21(𝑡)2𝑢𝑡22+𝐶𝑝𝜅2Ω(𝑢)𝑑𝑥,(2.13) where 𝐶𝑝 is the Poincaré constant, we have 1𝐿(𝑡)21+𝜆1+𝜆2𝑢𝑡22+121𝑡0(𝑠)𝑑𝑠+𝜆1𝐶𝑝𝑢22+𝑏(𝑡)𝑝+2𝑢𝑝+2𝑝+2+121+𝜆2𝐶𝑝𝜅Ω(𝑢)𝑑𝑥+𝑎(𝑡)2(𝑞+1)𝑢22(𝑞+1)+𝜆3Φ3(𝑡)+𝜆4Φ4(𝑡),𝑡0.(2.14) On the other hand, 2𝐿(𝑡)1𝜆1𝜆2𝑢𝑡22+1𝜆2𝐶𝑝𝜅Ω(𝑢)𝑑𝑥+2𝑏(𝑡)𝑝+2𝑢𝑝+2𝑝+2+𝑎(𝑡)𝑞+1𝑢22(𝑞+1)+1𝜅𝜆1𝐶𝑝𝑢22+2𝜆3Φ3(𝑡)+2𝜆4Φ4(𝑡).(2.15) Therefore, 𝜌1[(𝑡)+Φ3(𝑡)+Φ4(𝑡)]𝐿(𝑡)𝜌2[(𝑡)+Φ3(𝑡)+Φ4(𝑡)] for some constant 𝜌𝑖>0,𝑖=1,2 and small 𝜆𝑖,𝑖=1,2 such that 𝜆1<min{1,(1𝜅)/𝐶𝑝} and 𝜆2<min{1/𝐶𝑝𝜅,1𝜆1}.
The identity to follow is easy to justify and is helpful to prove our result.

Lemma 2.2. One has for 𝐶(0,) and 𝑣𝐶((0,);𝐿2(Ω))Ω𝑣(𝑡)𝑡01(𝑡𝑠)𝑣(𝑠)𝑑𝑠𝑑𝑥=2𝑡0(𝑠)𝑑𝑠𝑣(𝑡)22+12𝑡0(𝑡𝑠)𝑣(𝑠)221𝑑𝑠2Ω(𝑣)𝑑𝑥,𝑡0.(2.16)

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

Let 𝐼𝐑, and let 𝑔1,𝑔2𝐼𝐑{𝟎}. We write 𝑔1𝑔2 if 𝑔2/𝑔1 is nondecreasing in 𝐼.

Lemma 2.3. Let 𝑎(𝑡) be a positive continuous function in 𝐽=[𝛼,𝛽),𝑘𝑗(𝑡),𝑗=1,,𝑛 are nonnegative continuous functions, 𝑔𝑗(𝑢),𝑗=1,,𝑛 are nondecreasing continuous functions in 𝐑+, with 𝑔𝑗(𝑢)>0 for 𝑢>0, and 𝑢(𝑡) is a nonnegative continuous functions in 𝐽. If 𝑔1𝑔2𝑔𝑛 in (0,), then the inequality 𝑢(𝑡)𝑎(𝑡)+𝑛𝑗=1𝑡𝛼𝑘𝑗(𝑠)𝑔𝑗(𝑢(𝑠))𝑑𝑠,𝑡𝐽,(2.17) implies that 𝑢(𝑡)𝑐𝑛(𝑡),𝛼𝑡<𝛽0,(2.18) where 𝑐0(𝑡)=sup0𝑠𝑡𝑎(𝑠), 𝑐𝑗(𝑡)=𝐺𝑗1𝐺𝑗𝑐𝑗1(+𝑡)𝑡𝛼𝑘𝑗(𝐺𝑠)𝑑𝑠,𝑗=1,,𝑛,𝑗(𝑢)=𝑢𝑢𝑗𝑑𝑥𝑔𝑗(𝑢𝑥),𝑢>0𝑗,>0,𝑗=1,...,𝑛(2.19) and 𝛽0 is chosen so that the functions 𝑐𝑗(𝑡),𝑗=1,,𝑛 are defined for 𝛼𝑡<𝛽0.

Lemma 2.4. Assume that 2𝑞<+ if 𝑛=1,2 or 2𝑞<2𝑛/(𝑛2) if 𝑛3. Then there exists a positive constant 𝐶𝑒=𝐶𝑒(Ω,𝑞) such that 𝑢𝑞𝐶𝑒𝑢2(2.20) for 𝑢𝐻10(Ω).

3. Asymptotic Behavior

In this section we state and prove our result. To this end we need some notation. For every measurable set 𝒜𝐑+, we define the probability measure by 1(𝒜)=𝜅𝒜(𝑠)𝑑𝑠.(3.1) The nondecreasingness set and the non-decreasingness rate of are defined by 𝒬=𝑠𝐑+(𝑠)>0,(𝑠)0,(3.2)𝒬=,(3.3) respectively.

The following assumptions on the kernel (𝑡) will be adopted.(H1)(𝑡)0 for all 𝑡0 and 0<𝜅=0+(𝑠)𝑑𝑠<1.(H2) is absolutely continuous and of bounded variation on (0,) and (𝑡)𝜉(𝑡) for some nonnegative summable function 𝜉(𝑡) (=max{0,(𝑡)} where (𝑡) exists) and almost all 𝑡>0.(H3) There exists a nondecreasing function 𝛾(𝑡)>0 such that 𝛾(𝑡)/𝛾(𝑡)=𝜂(𝑡) is a nonincreasing function: 0+(𝑠)𝛾(𝑠)𝑑𝑠<+ and 0+𝜉(𝑠)𝛾(𝑠)𝑑𝑠<+.

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 𝑡>0 be a number such that 𝑡0(𝑠)𝑑𝑠=>0. We denote by 𝑡 the set 𝑡=[0,𝑡].

Lemma 3.1. One has for 𝑡𝑡 and 𝛿𝑖>0,𝑖=1,,5Φ2(𝑡)1𝛿1+32𝒬𝑡(𝑡𝑠)𝑑𝑠𝑢22+𝛿3𝑢𝑡22+1𝜅4𝛿1+11+𝛿2𝜅Ω𝒜𝑡||||(𝑡𝑠)𝑢(𝑡)𝑢(𝑠)2+1𝑑𝑠𝑑𝑥21𝒬𝑡(𝑡𝑠)𝑢(𝑠)22𝐶𝑑𝑠𝑝4𝛿3[]𝐵𝑉Ω+𝑢𝑑𝑥1+𝛿2𝒬𝑡(𝑡𝑠)𝑑𝑠Ω𝒬𝑡||||(𝑡𝑠)𝑢(𝑡)𝑢(𝑠)2+2𝑑𝑠𝑑𝑥𝑝+1𝛿4𝐶𝑒𝑏2(𝑡)(1𝜅)𝑝+1𝑝+1𝐶(𝑡)+𝑝𝜅4𝛿4Ω(𝑢)𝑑𝑥+𝛿5𝑎(𝑡)𝑢22(𝑞+1)+2𝑞1𝜅𝑎(𝑡)𝛿5(1𝜅)𝑞𝑞+1+𝐶(𝑡)𝑝4𝛿3𝒬𝑡𝜉(𝑡𝑠)𝑑𝑠Ω𝒬𝑡𝜉||||(𝑡𝑠)𝑢(𝑡)𝑢(𝑠)2𝑑𝑠𝑑𝑥,(3.4) where 𝐵𝑉[] is the total variation of .

Proof. This lemma is proved by a direct differentiation of Φ2(𝑡) along solutions of (1.1) and estimation of the different terms in the obtained expression of the derivative. Indeed, we have Φ2(𝑡)=Ω𝑢𝑡𝑡𝑡0(𝑡𝑠)(𝑢(𝑡)𝑢(𝑠))𝑑𝑠𝑑𝑥Ω𝑢𝑡𝑡0(𝑡𝑠)(𝑢(𝑡)𝑢(𝑠))𝑑𝑠+𝑢𝑡𝑡0(𝑠)𝑑𝑠𝑑𝑥(3.5) or Φ2(𝑡)=Ω1𝑡0(𝑠)𝑑𝑠Δ𝑢𝑏(𝑡)|𝑢|𝑝𝑢+𝑎(𝑡)𝑢22𝑞+Δ𝑢𝑡0(𝑡𝑠)(Δ𝑢(𝑡)Δ𝑢(𝑠))𝑑𝑠𝑡0(𝑡𝑠)(𝑢(𝑡)𝑢(𝑠))𝑑𝑠𝑑𝑥𝑡0𝑢(𝑠)𝑑𝑠𝑡22Ω𝑢𝑡𝑡0(𝑡𝑠)(𝑢(𝑡)𝑢(𝑠))𝑑𝑠𝑑𝑥,𝑡0.(3.6) Therefore, Φ2(𝑡)=1𝑡0(𝑠)𝑑𝑠+𝑎(𝑡)𝑢22𝑞×Ω𝑢𝑡0(𝑡𝑠)(𝑢(𝑡)𝑢(𝑠))𝑑𝑠𝑑𝑥𝑡0𝑢(𝑠)𝑑𝑠𝑡22+𝑏(𝑡)Ω|𝑢|𝑝𝑢𝑡0(𝑡𝑠)(𝑢(𝑡)𝑢(𝑠))𝑑𝑠𝑑𝑥Ω𝑢𝑡𝑡0+(𝑡𝑠)(𝑢(𝑡)𝑢(𝑠))𝑑𝑠𝑑𝑥Ω||||𝑡0||||(𝑡𝑠)(𝑢(𝑡)𝑢(𝑠))𝑑𝑠2𝑑𝑥,𝑡0.(3.7) For all measurable sets 𝒜 and 𝒬 such that 𝒜=𝐑+𝒬, it is clear that Ω𝑢𝑡0=(𝑡𝑠)(𝑢(𝑡)𝑢(𝑠))𝑑𝑠𝑑𝑥Ω𝑢𝒜[0,𝑡]+(𝑡𝑠)(𝑢(𝑡)𝑢(𝑠))𝑑𝑠𝑑𝑥Ω𝑢𝒬[0,𝑡]=(𝑡𝑠)(𝑢(𝑡)𝑢(𝑠))𝑑𝑠𝑑𝑥Ω𝑢𝒜[0,𝑡]+(𝑡𝑠)(𝑢(𝑡)𝑢(𝑠))𝑑𝑠𝑑𝑥𝒬[0,𝑡](𝑡𝑠)𝑑𝑠𝑢22Ω𝑢𝒬[0,𝑡](𝑡𝑠)𝑢(𝑠)𝑑𝑠𝑑𝑥,𝑡0.(3.8) For 𝛿1>0, the first term in the right-hand side of (3.8) satisfies Ω𝑢𝒜𝑡(𝑡𝑠)(𝑢(𝑡)𝑢(𝑠))𝑑𝑠𝑑𝑥𝛿1𝑢22+𝜅4𝛿1Ω𝒜𝑡||||(𝑡𝑠)𝑢(𝑡)𝑢(𝑠)2𝑑𝑠𝑑𝑥,𝑡0,(3.9) and the third one fulfills Ω𝑢𝒬𝑡1(𝑡𝑠)𝑢(𝑠)𝑑𝑠𝑑𝑥2𝒬𝑡(𝑡𝑠)𝑑𝑠𝑢22+12𝒬𝑡(𝑡𝑠)𝑢(𝑠)22𝑑𝑠,𝑡0.(3.10)
Back to (3.8) we may write Ω𝑢𝑡0(𝑡𝑠)(𝑢(𝑡)𝑢(𝑠))𝑑𝑠𝑑𝑥𝛿1𝑢22+𝜅4𝛿1Ω𝒜𝑡||||(𝑡𝑠)𝑢(𝑡)𝑢(𝑠)2+3𝑑𝑠𝑑𝑥2𝑢22𝒬𝑡1(𝑡𝑠)𝑑𝑠+2𝒬𝑡(𝑡𝑠)𝑢(𝑠)22𝑑𝑠,𝑡0.(3.11) The last term in the right-hand side of (3.7) will be estimated as follows: Ω||||𝑡0||||(𝑡𝑠)(𝑢(𝑡)𝑢(𝑠))𝑑s21𝑑𝑥1+𝛿2𝜅Ω𝒜𝑡||||(𝑡𝑠)𝑢(𝑡)𝑢(𝑠)2+𝑑𝑠𝑑𝑥1+𝛿2𝒬𝑡(𝑡𝑠)𝑑𝑠Ω𝒬𝑡||||(𝑡𝑠)𝑢(𝑡)𝑢(𝑠)2𝑑𝑠𝑑𝑥,𝛿2>0.(3.12) For the fourth term in (3.7), it holds that Ω𝑢𝑡𝑡0(𝑡𝑠)(𝑢(𝑡)𝑢(𝑠))𝑑𝑠𝑑𝑥𝛿3𝑢𝑡22𝐶𝑝4𝛿3[]𝐵𝑉Ω+𝐶𝑢𝑑𝑥𝑝4𝛿3𝒬𝑡𝜉(𝑡𝑠)𝑑𝑠Ω𝒬𝑡𝜉||||(𝑡𝑠)𝑢(𝑡)𝑢(𝑠)2𝑑𝑠𝑑𝑥,𝛿3>0,𝑡0.(3.13) Moreover, from Lemma 2.4, for 𝑝>0 if 𝑛=1,2 and 0<𝑝<2/(𝑛2) if 𝑛3, we find 𝑏(𝑡)Ω|𝑢|𝑝𝑢𝑡0(𝑡𝑠)(𝑢(𝑡)𝑢(𝑠))𝑑𝑠𝑑𝑥𝛿4𝑏2(𝑡)𝑢2(𝑝+1)2(𝑝+1)+𝐶𝑝4𝛿4𝑡0(𝑠)𝑑𝑠Ω(𝑢)𝑑𝑥𝛿4𝐶𝑒𝑏2(𝑡)𝑢22(𝑝+1)+𝐶𝑝𝜅4𝛿4Ω2(𝑢)𝑑𝑥𝑝+1𝛿4𝐶𝑒𝑏2(𝑡)(1𝜅)𝑝+1𝑝+1𝐶(𝑡)+𝑝𝜅4𝛿4Ω(𝑢)𝑑𝑥,𝛿4>0,𝑡0.(3.14) The definition of (𝑡) in (2.5) allows us to write 𝑎(𝑡)𝑢22𝑞Ω𝑢𝑡0(𝑡𝑠)(𝑢(𝑡)𝑢(𝑠))𝑑𝑠𝑑𝑥𝑎(𝑡)𝑢22𝑞𝛿5𝑢22+𝜅4𝛿5Ω(𝑢)𝑑𝑥𝛿5𝑎(𝑡)𝑢22(𝑞+1)+2𝑞1𝜅𝑎(𝑡)𝛿5(1𝜅)𝑞𝑞+1(𝑡),𝛿5>0,𝑡0.(3.15)
Gathering all the relations (3.11)–(3.15) together with (3.7), we obtain for 𝑡𝑡Φ2(𝑡)1𝛿1+32𝒬𝑡(𝑡𝑠)𝑑𝑠𝑢22+𝛿3𝑢𝑡22+1𝜅4𝛿1+11+𝛿2𝜅Ω𝒜𝑡||||(𝑡𝑠)𝑢(𝑡)𝑢(𝑠)2+1𝑑𝑠𝑑𝑥21𝒬𝑡(𝑡𝑠)𝑢(𝑠)22𝐶𝑑𝑠𝑝4𝛿3[]𝐵𝑉Ω+𝑢𝑑𝑥1+𝛿2𝒬𝑡(𝑡𝑠)𝑑𝑠Ω𝒬𝑡||||(𝑡𝑠)𝑢(𝑡)𝑢(𝑠)2+2𝑑𝑠𝑑𝑥𝑝+1𝛿4𝐶𝑒𝑏2(𝑡)(1𝜅)𝑝+1𝑝+1𝐶(𝑡)+𝑝𝜅4𝛿4Ω(𝑢)𝑑𝑥+𝛿5𝑎(𝑡)𝑢22(𝑞+1)+2𝑞1𝜅𝑎(𝑡)𝛿5(1𝜅)𝑞𝑞+1+𝐶(𝑡)𝑝4𝛿3𝒬𝑡𝜉(𝑡𝑠)𝑑𝑠Ω𝒬𝑡𝜉||||(𝑡𝑠)𝑢(𝑡)𝑢(𝑠)2𝑑𝑠𝑑𝑥.(3.16)

In the following theorem we will assume that 𝑝<𝑞 just to fix ideas. The result is also valid for 𝑝>𝑞. It suffices to interchange 𝑝𝑞 and 𝐴(𝑡)𝐵(𝑡) in the proof following it. The case 𝑝=𝑞 is easier.

We will make use of the following hypotheses for some positive constants 𝐴,𝐵,𝑈, and 𝑉 to be determined.(A)𝑎(𝑡) is a continuously differentiable function such that 𝑎(𝑡)<𝐴𝑎(𝑡),𝑡0.(B)𝑏(𝑡) is a continuously differentiable function such that 𝑏(𝑡)<𝐵𝑏(𝑡),𝑡0.(C)𝑝>0 if 𝑛=1,2 and 0<𝑝<2/(𝑛2) if 𝑛3.(D)[0𝑎(𝑠)𝑒𝑞𝑠𝑑𝑠]1/𝑞[0𝑏2(𝑠)𝑒𝑝𝑠𝑑𝑠]1/𝑝<𝑈. (E)[0𝑎(𝑠)𝑒𝑞𝑠0𝜂(𝜏)𝑑𝜏𝑑𝑠]1/𝑞[0𝑏2(𝑠)𝑒𝑝𝑠0𝜂(𝜏)𝑑𝜏𝑑𝑠]1/𝑝<𝑉.

Theorem 3.2. Assume that the hypotheses (H1)–(H3), (A)–(C) hold and <1/4. If lim𝑡𝜂(𝑡)=𝜂0, then, for global solutions and small 𝒬𝜉(𝑠)𝑑𝑠, there exist 𝑇1>0 and 𝑈>0 such that 𝐿(𝑡)>𝑈,𝑡𝑇1 or 𝐸(𝑡)𝑀1𝑒𝜈1𝑡,𝑡0(3.17) for some positive constants 𝑀1 and 𝜈1 as long as (D) holds. If 𝜂=0, then there exist 𝑇2>0 and 𝑉>0 such that 𝐿(𝑡)>𝑉,𝑡𝑇2 or 𝐸(𝑡)𝑀2𝑒𝜈2𝑡0𝜂(𝑠)𝑑𝑠,𝑡0(3.18) for some positive constants 𝑀2 and 𝜈2 as long as (E) holds.

Proof. A differentiation of Φ1(𝑡) with respect to 𝑡 along trajectories of (1.1) gives Φ1(𝑢𝑡)=𝑡22𝑢22+Ω𝑢𝑡0(𝑡𝑠)𝑢(𝑠)𝑑𝑠𝑑𝑥𝑎(𝑡)𝑢22(𝑞+1)𝑏(𝑡)𝑢𝑝+2𝑝+2,(3.19) and Lemma 2.2 implies Φ1(𝑢𝑡)𝑡22𝜅12𝑢22+12𝑡0(𝑡𝑠)𝑢(𝑠)221𝑑𝑠2Ω(𝑢)𝑑𝑥𝑎(𝑡)𝑢22(𝑞+1)𝑏(𝑡)𝑢𝑝+2𝑝+2,𝑡0.(3.20) Next, a differentiation of Φ3(𝑡) and Φ4(𝑡) yields Φ3(𝑡)=𝐻𝛾(0)𝑢22+𝑡0𝐻𝛾(𝑡𝑠)𝑢(𝑠)22𝑑𝑠=𝐻𝛾(0)𝑢22𝑡0𝛾(𝑡𝑠)𝐻𝛾(𝑡𝑠)𝛾(𝑡𝑠)𝑢(𝑠)22𝑑𝑠𝑡0(𝑡𝑠)𝑢(𝑠)22𝑑𝑠𝐻𝛾(0)𝑢22𝜂(𝑡)Φ3(𝑡)𝑡0(𝑡𝑠)𝑢(𝑠)22Φ𝑑𝑠,𝑡0,4(𝑡)=Ψ𝛾(0)𝑢22+𝑡0Ψ𝛾(𝑡𝑠)𝑢(𝑠)22𝑑𝑠=Ψ𝛾(0)𝑢22𝑡0𝛾(𝑡𝑠)Ψ𝛾(𝑡𝑠)𝛾(𝑡𝑠)𝑢(𝑠)22𝑑𝑠𝑡0𝜉(𝑡𝑠)𝑢(𝑠)22𝑑𝑠Ψ𝛾(0)𝑢22𝜂(𝑡)Φ4(𝑡)𝑡0𝜉(𝑡𝑠)𝑢(𝑠)22𝑑𝑠,𝑡0.(3.21)
Taking into account Lemma 3.1 and the relations (2.6), (3.20)-(3.21), we see that 𝐿(1𝑡)2𝐶𝑝4𝛿3𝜆2[]𝐵𝑉Ω𝜆𝑢𝑑𝑥+1+𝛿3𝜆2𝑢𝑡22+𝜆21𝛿1+32𝒬𝑡(𝑡𝑠)𝑑𝑠+𝜆3𝐻𝛾(0)+𝜆4Ψ𝛾(0)𝜆1𝜅12×𝑢22+𝜆12+𝜆212𝜆3𝑡0(𝑡𝑠)𝑢(𝑠)22+𝑎𝑑𝑠(𝑡)2(𝑞+1)+𝛿5𝜆2𝑎(𝑡)𝜆1𝑎(𝑡)𝑢22(𝑞+1)+1+𝛿2𝜆2𝒬𝑡𝜆(𝑡𝑠)𝑑𝑠12×Ω(𝑢)𝑑𝑥+𝜆2𝜅1+14𝛿1+1𝛿2Ω𝒜𝑡||||(𝑡𝑠)𝑢(𝑡)𝑢(𝑠)2𝑑𝑠𝑑𝑥𝜆3𝜂(𝑡)Φ3(𝑡)𝜆4𝜂(𝑡)Φ4𝑏(𝑡)+(𝑡)𝑝+2𝜆1𝑏(𝑡)𝑢𝑝+2𝑝+2+𝜆2𝐶𝑝𝜅4𝛿4Ω+2(𝑢)𝑑𝑥𝑞1𝜅𝑎(𝑡)𝛿5(1𝜅)𝑞𝜆2𝑞+12(𝑡)+𝑝+1𝛿4𝐶𝑒𝑏2(𝑡)(1𝜅)𝑝+1𝜆2𝑝+1(𝑡)𝜆4𝑡0𝜉(𝑡𝑠)𝑢(𝑠)22+𝜆𝑑𝑠2𝐶𝑝4𝛿3𝒬𝑡𝜉(𝑡𝑠)𝑑𝑠𝒬𝑡𝜉(𝑡𝑠)𝑢(𝑡)𝑢(𝑠)22𝑑𝑠,𝑡𝑡.(3.22) Next, as in [17], we introduce the sets 𝒜𝑛=𝑠𝐑+𝑛(𝑠)+(𝑠)0,𝑛𝐍,(3.23) and observe that 𝑛𝒜𝑛=𝐑+𝒬𝒩,(3.24) where 𝒩 is the null set where is not defined and 𝒬 is as in (3.2). Furthermore, if we denote 𝒬𝑛=𝐑+𝒜𝑛, then lim𝑛(𝒬𝑛)=(𝒬) because 𝒬𝑛+1𝒬𝑛 for all 𝑛 and 𝑛𝒬𝑛=𝒬𝒩. Moreover, we designate by 𝐴𝑛𝑡 the sets 𝐴𝑛𝑡=𝑠𝐑+0𝑠𝑡,𝑛(𝑡𝑠)+(𝑡𝑠)0,𝑛𝐍.(3.25) In (3.22), we take 𝒜𝑡𝐴=𝑛𝑡 and 𝒬𝑡𝒬=𝑛𝑡. Choosing 𝜆1=(𝜀)𝜆2, it is clear that 1+𝛿2𝜆2𝜅𝒬𝑛𝜆120(3.26) for small 𝜀 and 𝛿2, large 𝑛 and 𝑡, if (𝒬)<1/4. We deduce that 1+𝛿2𝜆2𝒬𝑛𝑡𝜆(𝑡𝑠)𝑑𝑠12<0.(3.27) Furthermore, if (𝒬)<1/4, then 312𝒬𝑛𝑡(𝑡𝑠)𝑑𝑠<𝛿𝜅12(3.28) with 3𝛿=1𝜅4(2𝜅)+𝛽(3.29) and a small 𝛽>0. Pick 𝜆3=12𝜆1+𝜆21(3.30) and 𝐻𝛾(0) such that 𝜆3𝐻𝛾(0)<𝜆2(1𝛿)(2𝜅)2.(3.31) Note that this is possible if 𝑡 is so large that >7𝜅/(8𝜅) even though 𝐻𝛾(0)=𝛾(0)10(𝑠)𝛾(𝑠)𝑑𝑠0(𝑠)𝑑𝑠=𝜅.(3.32)
Taking the relations (3.22)–(3.30) into account and selecting 𝜆2<𝛿3/𝐶𝑝𝐵𝑉[] so that 12𝐶𝑝𝜆24𝛿3[]1𝐵𝑉4,(3.33) and small enough so that 𝜆2𝜅1+14𝛿1+1𝛿2+𝐶𝑝4𝛿4<1,4𝑛(3.34) we find for 𝛿3=𝜀/2, large 𝛿4, small Ψ𝛾(0), and 𝑡𝑡𝐿(𝑡)𝐶1𝑢𝑡22+𝑢22+Ω+2(𝑢)𝑑𝑥𝑝+1𝛿4𝐶𝑒𝑏2(𝑡)(1𝜅)𝑝+1𝜆2𝑝+1+𝑎(𝑡)(𝑡)2(𝑞+1)+𝛿5𝜆2𝑎(𝑡)𝜆1𝑎(𝑡)𝑢22(𝑞+1)+2𝑞1𝜅𝑎(𝑡)𝛿5(1𝜅)𝑞𝜆2𝑞+1(𝑡)𝜆3𝜂(𝑡)Φ3(𝑡)𝜆4𝜂(𝑡)Φ4(1𝑡)+2𝜆1+2𝐶𝑝2𝛿3𝒬𝑡𝜉(𝑡𝑠)𝑑𝑠Ω𝒬𝑛𝑡(||||𝑡𝑠)𝑢(𝑡)𝑢(𝑠)2+𝑏𝑑𝑠𝑑𝑥(𝑡)𝑝+2𝜆1𝑏(𝑡)𝑢𝑝+2𝑝+2𝜆4𝑡0𝜉(𝑡𝑠)𝑢(𝑠)22𝑑𝑠(3.35) for some positive constant 𝐶1. Take 𝜆4>1+(𝜆2𝐶𝑝/2𝛿3)𝒬𝜉(𝑠)𝑑𝑠,𝛿5,𝒬𝜉(𝑠)𝑑𝑠 small, and 𝑎(𝑡)<𝜆2(𝑞+1)1𝛼𝑎(𝑡)(3.36) (i.e., 𝐴=2(𝑞+1)(𝜆1𝛼) for some 0<𝛼<𝜆1) and 𝑏(𝑡)<𝜆𝑝+21𝑏𝛽(𝑡)(3.37) (i.e., 𝐵=(𝑝+2)(𝜆1𝛽) for some 0<𝛽<𝜆1) to derive that 𝐿(𝑡)𝐶22(𝑡)+𝑝+1𝛿4𝐶𝑒𝑏2(𝑡)(1𝜅)𝑝+1𝜆2𝑝+12(𝑡)+𝑞1𝜅𝑎(𝑡)𝛿5(1𝜅)𝑞𝜆2𝑞+1(𝑡)𝜆3𝜂(𝑡)Φ3(𝑡)𝜆4𝜂(𝑡)Φ4(𝑡)(3.38) for some positive constant 𝐶2.
If lim𝑡𝜂(𝑡)0, then there exist a ̂𝑡𝑡 and 𝐶3>0 such that 𝜂(𝑡)𝐶3 for ̂𝑡𝑡. Thus, in virtue of Proposition 2.1, for 𝐶3>0, we have 𝐿(𝑡)𝐶3𝐿(𝑡)+𝐵(𝑡)𝐿𝑝+1(𝑡)+𝐴(𝑡)𝐿𝑞+1(𝑡),(3.39) where 2𝐵(𝑡)=𝑝+1𝛿4𝐶𝑒𝜆2(1𝜅)𝑝+1𝜌1𝑝+1𝑏22(𝑡),𝐴(𝑡)=𝑞1𝜅𝜆2𝛿5(1𝜅)𝑞𝜌1𝑞+1𝑎(𝑡).(3.40) If there exists a ̂𝑡𝑇 such that 𝐿(𝑇)<[𝑝0𝐵(𝑠)𝑑𝑠]1/𝑝, where 𝐵(𝑡)=𝐵(𝑡)𝑒𝑝𝐶3(𝑡𝑇), then from (3.39) 𝑒𝐶3(𝑡𝑇)𝐿(𝑡)𝑒𝐶3(𝑡𝑇)𝐵(𝑡)𝐿𝑝+1(𝑡)+𝑒𝐶3(𝑡𝑇)𝐴(𝑡)𝐿𝑞+1(𝑡),(3.41) and it follows that for 𝑡𝑇𝐿(𝑡)𝐿(𝑇)+𝑡𝑇𝑒𝑝𝐶3(𝑠𝑇)𝐿𝐵(𝑠)𝑝+1(𝑠)𝑑𝑠+𝑡𝑇𝑒𝑞𝐶3(𝑠𝑇)𝐿𝐴(𝑠)𝑞+1(𝑠)𝑑𝑠(3.42) with 𝐿(𝑡)=𝑒𝐶3(𝑡𝑇)𝐿(𝑡). Now we apply Lemma 2.3 to get 𝐿𝑁(𝑡)𝑞𝑞𝑡0𝐴(𝑠)𝑑𝑠1/𝑞,𝑡𝑇,(3.43) where 𝐴(𝑡)=𝑒𝑞𝐶3(𝑡𝑇)𝐴(𝑡) and 𝑁=[𝐿(𝑇)𝑝𝑝0𝐵(𝑠)𝑑𝑠]1/𝑝. If, in addition, 𝑞0𝐴(𝑠)𝑑𝑠<𝐿(𝑇)𝑞, then 𝐿(𝑡) is uniformly bounded by a positive constant 𝐶4. Thus 𝐿(𝑡)𝐶4𝑒𝐶3𝑡,𝑡𝑇,(3.44) and by continuity (3.44) holds for all 𝑡0.
If lim𝑡𝜂(𝑡)=0, then for any 𝐶>0 there exists a 𝑡(𝐶)𝑡 such that 𝜂(𝑡)𝐶 for 𝑡𝑡(𝐶). Therefore, 𝐿(𝑡)𝐶5𝜂(𝑡)𝐿(𝑡)+𝐵(𝑡)𝐿𝑝+1(𝑡)+𝐴(𝑡)𝐿𝑞+1(𝑡),𝑡𝑡=𝑡𝐶2(3.45) for some 𝐶5>0. The previous argument carries out with 𝑒𝐶3(𝑡𝑇) replaced by 𝑒𝐶5𝑡𝑇𝜂(𝑑)𝑑𝑠.
In case that 𝑞<𝑝, we reverse the roles of 𝑝 and 𝑞 in the argument above. The case 𝑝=𝑞 is clear.

Remark 3.3. The case where the derivative of the kernel does not approach zero on 𝒜 (as is the case, for instance, when 𝐶 on 𝒜) is interesting. Indeed, the right-hand side in condition (3.34) will be replaced by 𝐶/4 with a possibly large constant 𝐶.

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 𝑎(𝑡)<2(𝑞+1)(𝜆1𝛼)𝑎(𝑡) and 𝑏(𝑡)<(𝑝+2)(𝜆1𝛽)𝑏(𝑡) may be relaxed to 𝑎(𝑡)<2(𝑞+1)𝜆1𝑎(𝑡) and 𝑏(𝑡)<(𝑝+2)𝜆1𝑏(𝑡), respectively. In this case 𝛼=𝛼(𝑡) and 𝛽=𝛽(𝑡) would depend on 𝑡.

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.

References

  1. M. Aassila, M. M. Cavalcanti, and J. A. Soriano, “Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star-shaped domain,” SIAM Journal on Control and Optimization, vol. 38, no. 5, pp. 1581–1602, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. F. Alabau-Boussouira and P. Cannarsa, “A general method for proving sharp energy decay rates for memory-dissipative evolution equations,” Comptes Rendus Mathématique, vol. 347, no. 15-16, pp. 867–872, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. J. A. D. Appleby, M. Fabrizio, B. Lazzari, and D. W. Reynolds, “On exponential asymptotic stability in linear viscoelasticity,” Mathematical Models & Methods in Applied Sciences, vol. 16, no. 10, pp. 1677–1694, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. M. M. Cavalcanti, V. N. Domingos Cavalcanti, and P. Martinez, “General decay rate estimates for viscoelastic dissipative systems,” Nonlinear Analysis, vol. 68, no. 1, pp. 177–193, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. M. Fabrizio and S. Polidoro, “Asymptotic decay for some differential systems with fading memory,” Applicable Analysis, vol. 81, no. 6, pp. 1245–1264, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. K. M. Furati and N.-E. Tatar, “Uniform boundedness and stability for a viscoelastic problem,” Applied Mathematics and Computation, vol. 167, no. 2, pp. 1211–1220, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. X. S. Han and M. X. Wang, “Global existence and uniform decay for a nonlinear viscoelastic equation with damping,” Nonlinear Analysis, vol. 70, no. 9, pp. 3090–3098, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. W. Liu, “Uniform decay of solutions for a quasilinear system of viscoelastic equations,” Nonlinear Analysis, vol. 71, no. 5-6, pp. 2257–2267, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. W. Liu, “General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms,” Journal of Mathematical Physics, vol. 50, no. 11, Article ID 113506, 17 pages, 2009. View at Publisher · View at Google Scholar
  10. M. Medjden and N.-E. Tatar, “On the wave equation with a temporal non-local term,” Dynamic Systems and Applications, vol. 16, no. 4, pp. 665–671, 2007. View at Google Scholar · View at Zentralblatt MATH
  11. M. Medjden and N.-E. Tatar, “Asymptotic behavior for a viscoelastic problem with not necessarily decreasing kernel,” Applied Mathematics and Computation, vol. 167, no. 2, pp. 1221–1235, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. S. A. Messaoudi, “General decay of solutions of a viscoelastic equation,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 1457–1467, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. S. A. Messaoudi and N.-E. Tatar, “Global existence and uniform stability of solutions for a quasilinear viscoelastic problem,” Mathematical Methods in the Applied Sciences, vol. 30, no. 6, pp. 665–680, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. S. A. Messaoudi and N.-E. Tatar, “Exponential and polynomial decay for a quasilinear viscoelastic equation,” Nonlinear Analysis, vol. 68, no. 4, pp. 785–793, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. S. A. Messaoudi and N.-E. Tatar, “Exponential decay for a quasilinear viscoelastic equation,” Mathematische Nachrichten, vol. 282, no. 10, pp. 1443–1450, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. J. E. Muñoz Rivera and M. G. Naso, “On the decay of the energy for systems with memory and indefinite dissipation,” Asymptotic Analysis, vol. 49, no. 3-4, pp. 189–204, 2006. View at Google Scholar · View at Zentralblatt MATH
  17. V. Pata, “Exponential stability in linear viscoelasticity,” Quarterly of Applied Mathematics, vol. 64, no. 3, pp. 499–513, 2006. View at Google Scholar
  18. N.-E. Tatar, “On a problem arising in isothermal viscoelasticity,” International Journal of Pure and Applied Mathematics, vol. 8, no. 1, pp. 1–12, 2003. View at Google Scholar · View at Zentralblatt MATH
  19. N.-E. Tatar, “Long time behavior for a viscoelastic problem with a positive definite kernel,” The Australian Journal of Mathematical Analysis and Applications, vol. 1, no. 1, article 5, pp. 1–11, 2004. View at Google Scholar · View at Zentralblatt MATH
  20. N.-E. Tatar, “Polynomial stability without polynomial decay of the relaxation function,” Mathematical Methods in the Applied Sciences, vol. 31, no. 15, pp. 1874–1886, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. N.-E. Tatar, “How far can relaxation functions be increasing in viscoelastic problems?” Applied Mathematics Letters, vol. 22, no. 3, pp. 336–340, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. N.-E. Tatar, “Exponential decay for a viscoelastic problem with a singular kernel,” Zeitschrift für Angewandte Mathematik und Physik, vol. 60, no. 4, pp. 640–650, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. N.-E. Tatar, “On a large class of kernels yielding exponential stability in viscoelasticity,” Applied Mathematics and Computation, vol. 215, no. 6, pp. 2298–2306, 2009. View at Publisher · View at Google Scholar
  24. N.-E. Tatar, “Oscillating kernels and arbitrary decays in viscoelasticity,” Mathematische Nachrichten, vol. 285, no. 8-9, pp. 1130–1143, 2012. View at Publisher · View at Google Scholar
  25. S. Q. Yu, “Polynomial stability of solutions for a system of non-linear viscoelastic equations,” Applicable Analysis, vol. 88, no. 7, pp. 1039–1051, 2009. View at Publisher · View at Google Scholar
  26. M. Abdelli and A. Benaissa, “Energy decay of solutions of a degenerate Kirchhoff equation with a weak nonlinear dissipation,” Nonlinear Analysis, vol. 69, no. 7, pp. 1999–2008, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. S. Prates Filho, and J. A. Soriano, “Existence and exponential decay for a Kirchhoff-Carrier model with viscosity,” Journal of Mathematical Analysis and Applications, vol. 226, no. 1, pp. 40–60, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. A. Soriano, and J. S. Prates Filho, “Existence and asymptotic behaviour for a degenerate Kirchhoff-Carrier model with viscosity and nonlinear boundary conditions,” Revista Matemática Complutense, vol. 14, no. 1, pp. 177–203, 2001. View at Google Scholar · View at Zentralblatt MATH
  29. M. Hosoya and Y. Yamada, “On some nonlinear wave equations II: global existence and energy decay of solutions,” Journal of the Faculty of Science, The University of Tokyo IA, vol. 38, no. 1, pp. 239–250, 1991. View at Google Scholar · View at Zentralblatt MATH
  30. R. Ikehata, “A note on the global solvability of solutions to some nonlinear wave equations with dissipative terms,” Differential and Integral Equations, vol. 8, no. 3, pp. 607–616, 1995. View at Google Scholar · View at Zentralblatt MATH
  31. L. A. Medeiros and M. A. Milla Miranda, “On a nonlinear wave equation with damping,” Revista Matemática de la Universidad Complutense de Madrid, vol. 3, no. 2-3, pp. 213–231, 1990. View at Google Scholar · View at Zentralblatt MATH
  32. J. E. Muñoz Rivera and F. P. Quispe Gómez, “Existence and decay in non linear viscoelasticity,” Bollettino della Unione Matematica Italiana B, vol. 6, no. 1, pp. 1–37, 2003. View at Google Scholar
  33. K. Ono, “Global solvability for degenerate Kirchhoff equations with weak dissipation,” Mathematica Japonica, vol. 50, no. 3, pp. 409–413, 1999. View at Google Scholar · View at Zentralblatt MATH
  34. K. Ono, “On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation,” Mathematical Methods in the Applied Sciences, vol. 20, no. 2, pp. 151–177, 1997. View at Google Scholar
  35. G. A. P. Menzala, “On global classical solutions of a nonlinear wave equation,” Applicable Analysis, vol. 10, no. 3, pp. 179–195, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  36. A. Arosio and S. Garavaldi, “On the mildly degenerate Kirchhoff string,” Mathematical Methods in the Applied Sciences, vol. 14, no. 3, pp. 177–195, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  37. T. Kobayashi, H. Pecher, and Y. Shibata, “On a global in time existence theorem of smooth solutions to a nonlinear wave equation with viscosity,” Mathematische Annalen, vol. 296, no. 2, pp. 215–234, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  38. L. A. Medeiros, J. Limaco, and C. L. Frota, “On wave equations without global a priori estimates,” Boletim da Sociedade Paranaense de Matemática 3, vol. 30, no. 2, pp. 19–32, 2012. View at Google Scholar
  39. S. Mimouni, A. Benaissa, and N.-E. Amroun, “Global existence and optimal decay rate of solutions for the degenerate quasilinear wave equation with a strong dissipation,” Applicable Analysis, vol. 89, no. 6, pp. 815–831, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  40. J. E. Muñoz Rivera, “Global solution on a quasilinear wave equation with memory,” Bollettino della Unione Matematica Italiana B, vol. 8, no. 2, pp. 289–303, 1994. View at Google Scholar · View at Zentralblatt MATH
  41. K. Narashima, “Nonlinear vibration of an elastic string,” Journal of Sound and Vibration, vol. 8, no. 1, pp. 134–146, 1968. View at Google Scholar
  42. K. Ono, “Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings,” Journal of Differential Equations, vol. 137, no. 2, pp. 273–301, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  43. K. Ono, “On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation,” Journal of Mathematical Analysis and Applications, vol. 216, no. 1, pp. 321–342, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  44. J.-Y. Park and J.-J. Bae, “On the existence of solutions of strongly damped nonlinear wave equations,” International Journal of Mathematics and Mathematical Sciences, vol. 23, no. 6, pp. 369–382, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  45. R. M. Torrejón and J. M. Yong, “On a quasilinear wave equation with memory,” Nonlinear Analysis, vol. 16, no. 1, pp. 61–78, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  46. S.-T. Wu and L.-Y. Tsai, “On global existence and blow-up of solutions for an integro-differential equation with strong damping,” Taiwanese Journal of Mathematics, vol. 10, no. 4, pp. 979–1014, 2006. View at Google Scholar · View at Zentralblatt MATH
  47. M. Pinto, “Integral inequalities of Bihari-type and applications,” Funkcialaj Ekvacioj, vol. 33, no. 3, pp. 387–403, 1990. View at Google Scholar · View at Zentralblatt MATH