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Journal of Applied Mathematics
VolumeΒ 2012, Article IDΒ 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: YongkunΒ Li

Copyright Β© 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: 𝑒𝑑𝑑+π‘šξ“π‘–=1𝑏𝑖(𝑑)|𝑒|𝑝𝑖𝑒=1+π‘˜ξ“π‘—=1π‘Žπ‘—(𝑑)β€–βˆ‡π‘’β€–2π‘žπ‘—2ξƒͺβˆ’ξ€œΞ”π‘’π‘‘0β„Ž(π‘‘βˆ’π‘ )Δ𝑒(𝑠)𝑑𝑠,inΩ×𝐑+,𝑒=0,onΓ×𝐑+,𝑒(π‘₯,0)=𝑒0(π‘₯),𝑒𝑑(π‘₯,0)=𝑒1(π‘₯),inΞ©,(1.1) where Ξ© is a bounded domain in 𝐑𝑛 with smooth boundary Ξ“=πœ•Ξ© and 𝑝𝑖,π‘žπ‘—>0,𝑖=1,…,π‘š,𝑗=1,…,π‘˜. The functions 𝑒0(π‘₯) and 𝑒1(π‘₯) are given initial data, and the (nonnegative) functions π‘Žπ‘—(𝑑),𝑏𝑖(𝑑), and β„Ž(𝑑) 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 β„Ž. 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 π‘Žπ‘—(𝑑) and 𝑏𝑖(𝑑) are not nonincreasing, then we are in a nondissipative situation. This is the case also when the relaxation function oscillates (in case π‘Žπ‘—(𝑑),𝑏𝑖(𝑑) 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 π‘ˆ after which (the modified energy of) global solutions are bounded below by π‘ˆ 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,…,π‘š and π‘ž1=π‘ž,π‘žπ‘—=0,π‘Ž1=π‘Ž,π‘Žπ‘—=0,𝑗=2,…,π‘˜.

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

Theorem 1.1. Assume that (𝑒0,𝑒1)∈𝐻10(Ξ©)×𝐿2(Ξ©) and β„Ž(𝑑) is a nonnegative summable kernel. If 0<𝑝<2/(π‘›βˆ’2) when 𝑛β‰₯3 and 𝑝>0 when 𝑛=1,2, then there exists a unique solution 𝑒 to problem (1.1) such that ξ€·[]π‘’βˆˆπΆ0,𝑇;𝐻10ξ€Έ(Ξ©)∩𝐢1ξ€·[]0,𝑇;𝐿2ξ€Έ(Ξ©)(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(𝑑)=2‖‖𝑒𝑑‖‖22+β€–βˆ‡π‘’β€–22+π‘Ž(𝑑)β€–2(π‘ž+1)βˆ‡π‘’β€–22(π‘ž+1)+𝑏(𝑑)‖𝑝+2𝑒‖𝑝+2𝑝+2,𝑑β‰₯0,(2.1) where ‖⋅‖𝑝 denotes the norm in 𝐿𝑝(Ξ©). Then by  (1.1)  it is easy to see that for 𝑑β‰₯0πΈξ…ž(ξ€œπ‘‘)=Ξ©βˆ‡π‘’π‘‘β‹…ξ€œπ‘‘0π‘Žβ„Ž(π‘‘βˆ’π‘ )βˆ‡π‘’(𝑠)𝑑𝑠𝑑π‘₯+ξ…ž(𝑑)2(π‘ž+1)β€–βˆ‡π‘’β€–22(π‘ž+1)+π‘ξ…ž(𝑑)𝑝+2‖𝑒‖𝑝+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, ξ€œΞ©βˆ‡π‘’π‘‘β‹…ξ€œπ‘‘01β„Ž(π‘‘βˆ’π‘ )βˆ‡π‘’(𝑠)𝑑𝑠𝑑π‘₯=2ξ€œΞ©ξ€·β„Žξ…žξ€Έ1β–‘βˆ‡π‘’π‘‘π‘₯βˆ’2β„Ž(𝑑)β€–βˆ‡π‘’β€–22βˆ’12π‘‘ξ‚»ξ€œπ‘‘π‘‘Ξ©ξ‚΅ξ€œ(β„Žβ–‘βˆ‡π‘’)𝑑π‘₯βˆ’π‘‘0ξ‚Άβ„Ž(𝑠)π‘‘π‘ β€–βˆ‡π‘’β€–22ξ‚Ό,(2.3) where (ξ€œβ„Žβ–‘π‘£)(𝑑)∢=𝑑0||||β„Ž(π‘‘βˆ’π‘ )𝑣(𝑑)βˆ’π‘£(𝑠)2𝑑𝑠.(2.4) Therefore, if we modify 𝐸(𝑑) to 1β„°(𝑑)∢=2‖‖𝑒𝑑‖‖22+ξ‚΅ξ€œ1βˆ’π‘‘0ξ‚Άβ„Ž(𝑠)π‘‘π‘ β€–βˆ‡π‘’β€–22+ξ€œΞ©(ξ‚Ό+β„Žβ–‘βˆ‡π‘’)𝑑π‘₯π‘Ž(𝑑)β€–2(π‘ž+1)βˆ‡π‘’β€–22(π‘ž+1)+𝑏(𝑑)‖𝑝+2𝑒‖𝑝+2𝑝+2,(2.5) we obtain for 𝑑β‰₯0β„°ξ…ž1(𝑑)=2ξ€œΞ©ξ‚€ξ€·β„Žξ…žξ€Έ||||β–‘βˆ‡π‘’βˆ’β„Ž(𝑑)βˆ‡π‘’2ξ‚π‘Žπ‘‘π‘₯+ξ…ž(𝑑)2(π‘ž+1)β€–βˆ‡π‘’β€–22(π‘ž+1)+π‘ξ…ž(𝑑)𝑝+2‖𝑒‖𝑝+2𝑝+2.(2.6) Assuming that ξ€œ1βˆ’0+βˆžβ„Ž(𝑠)𝑑𝑠=∢1βˆ’πœ…>0(2.7) makes β„°(𝑑) a nonnegative functional. The following functionals Ξ¦1(ξ€œπ‘‘)∢=Ω𝑒𝑑Φ𝑒𝑑π‘₯,2ξ€œ(𝑑)∢=βˆ’Ξ©π‘’π‘‘ξ€œπ‘‘0β„Ž(π‘‘βˆ’π‘ )(𝑒(𝑑)βˆ’π‘’(𝑠))𝑑𝑠𝑑π‘₯(2.8) are standard and will be used here. The next ones have been introduced by the present author in [24] Ξ¦3(ξ€œπ‘‘)∢=𝑑0𝐻𝛾(π‘‘βˆ’π‘ )β€–βˆ‡π‘’(𝑠)β€–22𝑑𝑠,Ξ¦4(ξ€œπ‘‘)∢=𝑑0Ψ𝛾(π‘‘βˆ’π‘ )β€–βˆ‡π‘’(𝑠)β€–22𝑑𝑠,(2.9) where 𝐻𝛾(𝑑)∢=𝛾(𝑑)βˆ’1ξ€œβˆžπ‘‘β„Ž(𝑠)𝛾(𝑠)𝑑𝑠,Ψ𝛾(𝑑)∢=𝛾(𝑑)βˆ’1ξ€œβˆžπ‘‘πœ‰(𝑠)𝛾(𝑠)𝑑𝑠,𝑑β‰₯0,(2.10) and 𝛾(𝑑) and πœ‰(𝑑) are two nonnegative functions which will be precised later (see (H2), (H3)). The functional 𝐿(𝑑)∢=β„°(𝑑)+4𝑖=1πœ†π‘–Ξ¦π‘–(𝑑)(2.11) for some πœ†π‘–>0,𝑖=1,2,3,4, to be determined is equivalent to β„°(𝑑)+Ξ¦3(𝑑)+Ξ¦4(𝑑).

Proposition 2.1. There exist πœŒπ‘–>0,𝑖=1,2 such that 𝜌1ξ€Ίβ„°(𝑑)+Ξ¦3(𝑑)+Ξ¦4ξ€»(𝑑)≀𝐿(𝑑)β‰€πœŒ2ξ€Ίβ„°(𝑑)+Ξ¦3(𝑑)+Ξ¦4ξ€»(𝑑)(2.12) for all 𝑑β‰₯0 and small πœ†π‘–,𝑖=1,2.

Proof. By the inequalities Ξ¦1(ξ€œπ‘‘)=Ω𝑒𝑑1𝑒𝑑π‘₯≀2‖‖𝑒𝑑‖‖22+𝐢𝑝2β€–βˆ‡π‘’β€–22,Ξ¦21(𝑑)≀2‖‖𝑒𝑑‖‖22+πΆπ‘πœ…2ξ€œΞ©(β„Žβ–‘βˆ‡π‘’)𝑑π‘₯,(2.13) where 𝐢𝑝 is the PoincarΓ© constant, we have 1𝐿(𝑑)≀2ξ€·1+πœ†1+πœ†2‖‖𝑒𝑑‖‖22+12ξ‚΅ξ€œ1βˆ’π‘‘0β„Ž(𝑠)𝑑𝑠+πœ†1πΆπ‘ξ‚Άβ€–βˆ‡π‘’β€–22+𝑏(𝑑)𝑝+2‖𝑒‖𝑝+2𝑝+2+12ξ€·1+πœ†2πΆπ‘πœ…ξ€Έξ€œΞ©(β„Žβ–‘βˆ‡π‘’)𝑑π‘₯+π‘Ž(𝑑)2(π‘ž+1)β€–βˆ‡π‘’β€–22(π‘ž+1)+πœ†3Ξ¦3(𝑑)+πœ†4Ξ¦4(𝑑),𝑑β‰₯0.(2.14) On the other hand, ξ€·2𝐿(𝑑)β‰₯1βˆ’πœ†1βˆ’πœ†2‖‖𝑒𝑑‖‖22+ξ€·1βˆ’πœ†2πΆπ‘πœ…ξ€Έξ€œΞ©(β„Žβ–‘βˆ‡π‘’)𝑑π‘₯+2𝑏(𝑑)𝑝+2‖𝑒‖𝑝+2𝑝+2+π‘Ž(𝑑)π‘ž+1β€–βˆ‡π‘’β€–22(π‘ž+1)+ξ€Ί1βˆ’πœ…βˆ’πœ†1πΆπ‘ξ€»β€–βˆ‡π‘’β€–22+2πœ†3Ξ¦3(𝑑)+2πœ†4Ξ¦4(𝑑).(2.15) Therefore, 𝜌1[β„°(𝑑)+Ξ¦3(𝑑)+Ξ¦4(𝑑)]≀𝐿(𝑑)β‰€πœŒ2[β„°(𝑑)+Ξ¦3(𝑑)+Ξ¦4(𝑑)] for some constant πœŒπ‘–>0,𝑖=1,2 and small πœ†π‘–,𝑖=1,2 such that πœ†1<min{1,(1βˆ’πœ…)/𝐢𝑝} and πœ†2<min{1/πΆπ‘πœ…,1βˆ’πœ†1}.
The identity to follow is easy to justify and is helpful to prove our result.

Lemma 2.2. One has for β„ŽβˆˆπΆ(0,∞) and π‘£βˆˆπΆ((0,∞);𝐿2(Ξ©))ξ€œΞ©ξ€œπ‘£(𝑑)𝑑01β„Ž(π‘‘βˆ’π‘ )𝑣(𝑠)𝑑𝑠𝑑π‘₯=2ξ‚΅ξ€œπ‘‘0ξ‚Άβ„Ž(𝑠)𝑑𝑠‖𝑣(𝑑)β€–22+12ξ€œπ‘‘0β„Ž(π‘‘βˆ’π‘ )‖𝑣(𝑠)β€–221π‘‘π‘ βˆ’2ξ€œΞ©(β„Žβ–‘π‘£)𝑑π‘₯,𝑑β‰₯0.(2.16)

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

Let πΌβŠ‚π‘, and let 𝑔1,𝑔2βˆΆπΌβ†’π‘β§΅{𝟎}. We write 𝑔1βˆπ‘”2 if 𝑔2/𝑔1 is nondecreasing in 𝐼.

Lemma 2.3. Let π‘Ž(𝑑) be a positive continuous function in 𝐽∢=[𝛼,𝛽),π‘˜π‘—(𝑑),𝑗=1,…,𝑛 are nonnegative continuous functions, 𝑔𝑗(𝑒),𝑗=1,…,𝑛 are nondecreasing continuous functions in 𝐑+, with 𝑔𝑗(𝑒)>0 for 𝑒>0, and 𝑒(𝑑) is a nonnegative continuous functions in 𝐽. If 𝑔1βˆπ‘”2βˆβ‹―βˆπ‘”π‘› in (0,∞), then the inequality 𝑒(𝑑)β‰€π‘Ž(𝑑)+𝑛𝑗=1ξ€œπ‘‘π›Όπ‘˜π‘—(𝑠)𝑔𝑗(𝑒(𝑠))𝑑𝑠,π‘‘βˆˆπ½,(2.17) implies that 𝑒(𝑑)≀𝑐𝑛(𝑑),𝛼≀𝑑<𝛽0,(2.18) where 𝑐0(𝑑)∢=sup0β‰€π‘ β‰€π‘‘π‘Ž(𝑠), 𝑐𝑗(𝑑)∢=πΊπ‘—βˆ’1ξ‚ΈπΊπ‘—ξ€·π‘π‘—βˆ’1(ξ€Έ+ξ€œπ‘‘)π‘‘π›Όπ‘˜π‘—(𝐺𝑠)𝑑𝑠,𝑗=1,…,𝑛,π‘—ξ€œ(𝑒)∢=𝑒𝑒𝑗𝑑π‘₯𝑔𝑗(𝑒π‘₯),𝑒>0𝑗,>0,𝑗=1,...,𝑛(2.19) and 𝛽0 is chosen so that the functions 𝑐𝑗(𝑑),𝑗=1,…,𝑛 are defined for 𝛼≀𝑑<𝛽0.

Lemma 2.4. Assume that 2β‰€π‘ž<+∞ if 𝑛=1,2 or 2β‰€π‘ž<2𝑛/(π‘›βˆ’2) if 𝑛β‰₯3. Then there exists a positive constant 𝐢𝑒=𝐢𝑒(Ξ©,π‘ž) such that β€–π‘’β€–π‘žβ‰€πΆπ‘’β€–βˆ‡π‘’β€–2(2.20) for π‘’βˆˆπ»10(Ξ©).

3. Asymptotic Behavior

In this section we state and prove our result. To this end we need some notation. For every measurable set π’œβŠ‚π‘+, we define the probability measure ξβ„Ž by 1β„Ž(π’œ)∢=πœ…ξ€œπ’œβ„Ž(𝑠)𝑑𝑠.(3.1) The nondecreasingness set and the non-decreasingness rate of β„Ž are defined by π’¬β„Žξ€½βˆΆ=π‘ βˆˆπ‘+βˆΆβ„Ž(𝑠)>0,β„Žξ…žξ€Ύβ„›(𝑠)β‰₯0,(3.2)β„Žξβ„Žξ€·π’¬βˆΆ=β„Žξ€Έ,(3.3) respectively.

The following assumptions on the kernel β„Ž(𝑑) will be adopted.(H1)β„Ž(𝑑)β‰₯0 for all 𝑑β‰₯0 and ∫0<πœ…=0+βˆžβ„Ž(𝑠)𝑑𝑠<1.(H2)β„Ž is absolutely continuous and of bounded variation on (0,∞) and β„Žξ…ž(𝑑)β‰€πœ‰(𝑑) for some nonnegative summable function πœ‰(𝑑) (=max{0,β„Žξ…ž(𝑑)} where β„Žξ…ž(𝑑) exists) and almost all 𝑑>0.(H3) There exists a nondecreasing function 𝛾(𝑑)>0 such that π›Ύξ…ž(𝑑)/𝛾(𝑑)=πœ‚(𝑑) is a nonincreasing function: ∫0+βˆžβ„Ž(𝑠)𝛾(𝑠)𝑑𝑠<+∞ and ∫0+βˆžπœ‰(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 π‘‘βˆ—>0 be a number such that βˆ«π‘‘βˆ—0β„Ž(𝑠)𝑑𝑠=β„Žβˆ—>0. We denote by ℬ𝑑 the set β„¬π‘‘βˆΆ=β„¬βˆ©[0,𝑑].

Lemma 3.1. One has for 𝑑β‰₯π‘‘βˆ— and 𝛿𝑖>0,𝑖=1,…,5Ξ¦ξ…ž2ξ€·(𝑑)≀1βˆ’β„Žβˆ—ξ€Έξ‚Έπ›Ώ1+32ξ€œπ’¬π‘‘ξ‚Ήβ„Ž(π‘‘βˆ’π‘ )π‘‘π‘ β€–βˆ‡π‘’β€–22+𝛿3βˆ’β„Žβˆ—ξ€Έβ€–β€–π‘’π‘‘β€–β€–22+ξ‚Έξ€·1βˆ’β„Žβˆ—ξ€Έπœ…4𝛿1+ξ‚΅11+𝛿2ξ‚Άπœ…ξ‚Ήξ€œΞ©ξ€œπ’œπ‘‘||||β„Ž(π‘‘βˆ’π‘ )βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠)2+1𝑑𝑠𝑑π‘₯2ξ€·1βˆ’β„Žβˆ—ξ€Έξ€œπ’¬π‘‘β€–β„Ž(π‘‘βˆ’π‘ )β€–βˆ‡π‘’(𝑠)22πΆπ‘‘π‘ βˆ’π‘4𝛿3[β„Ž]ξ€œπ΅π‘‰Ξ©ξ€·β„Žξ…žξ€Έ+ξ€·β–‘βˆ‡π‘’π‘‘π‘₯1+𝛿2ξ€Έξ‚΅ξ€œπ’¬π‘‘ξ‚Άξ€œβ„Ž(π‘‘βˆ’π‘ )π‘‘π‘ Ξ©ξ€œπ’¬π‘‘||||β„Ž(π‘‘βˆ’π‘ )βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠)2+2𝑑𝑠𝑑π‘₯𝑝+1𝛿4𝐢𝑒𝑏2(𝑑)(1βˆ’πœ…)𝑝+1ℰ𝑝+1𝐢(𝑑)+π‘πœ…4𝛿4ξ€œΞ©(β„Žβ–‘βˆ‡π‘’)𝑑π‘₯+𝛿5π‘Ž(𝑑)β€–βˆ‡π‘’β€–22(π‘ž+1)+2π‘žβˆ’1πœ…π‘Ž(𝑑)𝛿5(1βˆ’πœ…)π‘žβ„°π‘ž+1+𝐢(𝑑)𝑝4𝛿3ξ‚΅ξ€œπ’¬π‘‘πœ‰ξ‚Άξ€œ(π‘‘βˆ’π‘ )π‘‘π‘ Ξ©ξ€œπ’¬π‘‘πœ‰||||(π‘‘βˆ’π‘ )βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠)2𝑑𝑠𝑑π‘₯,(3.4) where 𝐡𝑉[β„Ž] is the total variation of β„Ž.

Proof. This lemma is proved by a direct differentiation of Ξ¦2(𝑑) along solutions of (1.1) and estimation of the different terms in the obtained expression of the derivative. Indeed, we have Ξ¦ξ…ž2(ξ€œπ‘‘)=βˆ’Ξ©π‘’π‘‘π‘‘ξ€œπ‘‘0βˆ’ξ€œβ„Ž(π‘‘βˆ’π‘ )(𝑒(𝑑)βˆ’π‘’(𝑠))𝑑𝑠𝑑π‘₯Ξ©π‘’π‘‘ξ‚Έξ€œπ‘‘0β„Žξ…ž(π‘‘βˆ’π‘ )(𝑒(𝑑)βˆ’π‘’(𝑠))𝑑𝑠+π‘’π‘‘ξ€œπ‘‘0ξ‚Ήβ„Ž(𝑠)𝑑𝑠𝑑π‘₯(3.5) or Ξ¦ξ…ž2(ξ€œπ‘‘)=βˆ’Ξ©ξ€œξ‚Έξ‚΅1βˆ’π‘‘0ξ‚Άβ„Ž(𝑠)π‘‘π‘ Ξ”π‘’βˆ’π‘(𝑑)|𝑒|𝑝𝑒+π‘Ž(𝑑)β€–βˆ‡π‘’β€–22π‘ž+ξ€œΞ”π‘’π‘‘0ξ‚Ήξ€œβ„Ž(π‘‘βˆ’π‘ )(Δ𝑒(𝑑)βˆ’Ξ”π‘’(𝑠))𝑑𝑠𝑑0βˆ’ξ‚΅ξ€œβ„Ž(π‘‘βˆ’π‘ )(𝑒(𝑑)βˆ’π‘’(𝑠))𝑑𝑠𝑑π‘₯𝑑0ξ‚Άβ€–β€–π‘’β„Ž(𝑠)𝑑𝑠𝑑‖‖22βˆ’ξ€œΞ©π‘’π‘‘ξ€œπ‘‘0β„Žξ…ž(π‘‘βˆ’π‘ )(𝑒(𝑑)βˆ’π‘’(𝑠))𝑑𝑠𝑑π‘₯,𝑑β‰₯0.(3.6) Therefore, Ξ¦ξ…ž2(ξ‚΅ξ€œπ‘‘)=1βˆ’π‘‘0β„Ž(𝑠)𝑑𝑠+π‘Ž(𝑑)β€–βˆ‡π‘’β€–22π‘žξ‚ΆΓ—ξ€œΞ©ξ€œβˆ‡π‘’β‹…π‘‘0ξ‚΅ξ€œβ„Ž(π‘‘βˆ’π‘ )(βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠))𝑑𝑠𝑑π‘₯βˆ’π‘‘0ξ‚Άβ€–β€–π‘’β„Ž(𝑠)𝑑𝑠𝑑‖‖22ξ€œ+𝑏(𝑑)Ξ©|𝑒|π‘π‘’ξ€œπ‘‘0ξ€œβ„Ž(π‘‘βˆ’π‘ )(𝑒(𝑑)βˆ’π‘’(𝑠))𝑑𝑠𝑑π‘₯βˆ’Ξ©π‘’π‘‘ξ€œπ‘‘0β„Žξ…ž+ξ€œ(π‘‘βˆ’π‘ )(𝑒(𝑑)βˆ’π‘’(𝑠))𝑑𝑠𝑑π‘₯Ξ©||||ξ€œπ‘‘0||||β„Ž(π‘‘βˆ’π‘ )(βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠))𝑑𝑠2𝑑π‘₯,𝑑β‰₯0.(3.7) For all measurable sets π’œ and 𝒬 such that π’œ=𝐑+⧡𝒬, it is clear that ξ€œΞ©ξ€œβˆ‡π‘’β‹…π‘‘0=ξ€œβ„Ž(π‘‘βˆ’π‘ )(βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠))𝑑𝑠𝑑π‘₯Ξ©ξ€œβˆ‡π‘’β‹…π’œβˆ©[0,𝑑]+ξ€œβ„Ž(π‘‘βˆ’π‘ )(βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠))𝑑𝑠𝑑π‘₯Ξ©ξ€œβˆ‡π‘’β‹…π’¬βˆ©[0,𝑑]β„Ž=ξ€œ(π‘‘βˆ’π‘ )(βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠))𝑑𝑠𝑑π‘₯Ξ©ξ€œβˆ‡π‘’β‹…π’œβˆ©[0,𝑑]+ξ‚΅ξ€œβ„Ž(π‘‘βˆ’π‘ )(βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠))𝑑𝑠𝑑π‘₯π’¬βˆ©[0,𝑑]ξ‚Άβ„Ž(π‘‘βˆ’π‘ )π‘‘π‘ β€–βˆ‡π‘’β€–22βˆ’ξ€œΞ©ξ€œβˆ‡π‘’β‹…π’¬βˆ©[0,𝑑]β„Ž(π‘‘βˆ’π‘ )βˆ‡π‘’(𝑠)𝑑𝑠𝑑π‘₯,𝑑β‰₯0.(3.8) For 𝛿1>0, the first term in the right-hand side of (3.8) satisfies ξ€œΞ©ξ€œβˆ‡π‘’β‹…π’œπ‘‘β„Ž(π‘‘βˆ’π‘ )(βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠))𝑑𝑠𝑑π‘₯≀𝛿1β€–βˆ‡π‘’β€–22+πœ…4𝛿1ξ€œΞ©ξ€œπ’œπ‘‘||||β„Ž(π‘‘βˆ’π‘ )βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠)2𝑑𝑠𝑑π‘₯,𝑑β‰₯0,(3.9) and the third one fulfills ξ€œΞ©ξ€œβˆ‡π‘’β‹…π’¬π‘‘β‰€1β„Ž(π‘‘βˆ’π‘ )βˆ‡π‘’(𝑠)𝑑𝑠𝑑π‘₯2ξ‚΅ξ€œπ’¬π‘‘ξ‚Άβ„Ž(π‘‘βˆ’π‘ )π‘‘π‘ β€–βˆ‡π‘’β€–22+12ξ€œπ’¬π‘‘β„Ž(π‘‘βˆ’π‘ )β€–βˆ‡π‘’(𝑠)β€–22𝑑𝑠,𝑑β‰₯0.(3.10)
Back to (3.8) we may write ξ€œΞ©ξ€œβˆ‡π‘’β‹…π‘‘0β„Ž(π‘‘βˆ’π‘ )(βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠))𝑑𝑠𝑑π‘₯≀𝛿1β€–βˆ‡π‘’β€–22+πœ…4𝛿1ξ€œΞ©ξ€œπ’œπ‘‘||||β„Ž(π‘‘βˆ’π‘ )βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠)2+3𝑑𝑠𝑑π‘₯2β€–βˆ‡π‘’β€–22ξ€œπ’¬π‘‘1β„Ž(π‘‘βˆ’π‘ )𝑑𝑠+2ξ€œπ’¬π‘‘β„Ž(π‘‘βˆ’π‘ )β€–βˆ‡π‘’(𝑠)β€–22𝑑𝑠,𝑑β‰₯0.(3.11) The last term in the right-hand side of (3.7) will be estimated as follows: ξ€œΞ©||||ξ€œπ‘‘0||||β„Ž(π‘‘βˆ’π‘ )(βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠))𝑑s2≀1𝑑π‘₯1+𝛿2ξ‚Άπœ…ξ€œΞ©ξ€œπ’œπ‘‘||||β„Ž(π‘‘βˆ’π‘ )βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠)2+𝑑𝑠𝑑π‘₯1+𝛿2ξ€Έξ‚΅ξ€œπ’¬π‘‘β„Žξ‚Άξ€œ(π‘‘βˆ’π‘ )π‘‘π‘ Ξ©ξ€œπ’¬π‘‘β„Ž||||(π‘‘βˆ’π‘ )βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠)2𝑑𝑠𝑑π‘₯,𝛿2>0.(3.12) For the fourth term in (3.7), it holds that βˆ’ξ€œΞ©π‘’π‘‘ξ€œπ‘‘0β„Žξ…ž(π‘‘βˆ’π‘ )(𝑒(𝑑)βˆ’π‘’(𝑠))𝑑𝑠𝑑π‘₯≀𝛿3‖‖𝑒𝑑‖‖22βˆ’πΆπ‘4𝛿3[β„Ž]ξ€œπ΅π‘‰Ξ©ξ€·β„Žξ…žξ€Έ+πΆβ–‘βˆ‡π‘’π‘‘π‘₯𝑝4𝛿3ξ‚΅ξ€œπ’¬π‘‘πœ‰ξ‚Άξ€œ(π‘‘βˆ’π‘ )π‘‘π‘ Ξ©ξ€œπ’¬π‘‘πœ‰||||(π‘‘βˆ’π‘ )βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠)2𝑑𝑠𝑑π‘₯,𝛿3>0,𝑑β‰₯0.(3.13) Moreover, from Lemma 2.4, for 𝑝>0 if 𝑛=1,2 and 0<𝑝<2/(π‘›βˆ’2) if 𝑛β‰₯3, we find ξ€œπ‘(𝑑)Ξ©|𝑒|π‘π‘’ξ€œπ‘‘0β„Ž(π‘‘βˆ’π‘ )(𝑒(𝑑)βˆ’π‘’(𝑠))𝑑𝑠𝑑π‘₯≀𝛿4𝑏2(𝑑)‖𝑒‖2(𝑝+1)2(𝑝+1)+𝐢𝑝4𝛿4ξ‚΅ξ€œπ‘‘0ξ‚Άξ€œβ„Ž(𝑠)𝑑𝑠Ω(β„Žβ–‘βˆ‡π‘’)𝑑π‘₯≀𝛿4𝐢𝑒𝑏2(𝑑)β€–βˆ‡π‘’β€–22(𝑝+1)+πΆπ‘πœ…4𝛿4ξ€œΞ©β‰€2(β„Žβ–‘βˆ‡π‘’)𝑑π‘₯𝑝+1𝛿4𝐢𝑒𝑏2(𝑑)(1βˆ’πœ…)𝑝+1ℰ𝑝+1𝐢(𝑑)+π‘πœ…4𝛿4ξ€œΞ©(β„Žβ–‘βˆ‡π‘’)𝑑π‘₯,𝛿4>0,𝑑β‰₯0.(3.14) The definition of β„°(𝑑) in (2.5) allows us to write π‘Ž(𝑑)β€–βˆ‡π‘’β€–22π‘žξ€œΞ©ξ€œβˆ‡π‘’β‹…π‘‘0β„Ž(π‘‘βˆ’π‘ )(βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠))𝑑𝑠𝑑π‘₯β‰€π‘Ž(𝑑)β€–βˆ‡π‘’β€–22π‘žξ‚»π›Ώ5β€–βˆ‡π‘’β€–22+πœ…4𝛿5ξ€œΞ©(ξ‚Όβ„Žβ–‘βˆ‡π‘’)𝑑π‘₯≀𝛿5π‘Ž(𝑑)β€–βˆ‡π‘’β€–22(π‘ž+1)+2π‘žβˆ’1πœ…π‘Ž(𝑑)𝛿5(1βˆ’πœ…)π‘žβ„°π‘ž+1(𝑑),𝛿5>0,𝑑β‰₯0.(3.15)
Gathering all the relations (3.11)–(3.15) together with (3.7), we obtain for 𝑑β‰₯π‘‘βˆ—Ξ¦ξ…ž2ξ€·(𝑑)≀1βˆ’β„Žβˆ—ξ€Έξ‚Έπ›Ώ1+32ξ€œπ’¬π‘‘ξ‚Ήβ„Ž(π‘‘βˆ’π‘ )π‘‘π‘ β€–βˆ‡π‘’β€–22+𝛿3βˆ’β„Žβˆ—ξ€Έβ€–β€–π‘’π‘‘β€–β€–22+ξ‚Έξ€·1βˆ’β„Žβˆ—ξ€Έπœ…4𝛿1+ξ‚΅11+𝛿2ξ‚Άπœ…ξ‚Ήξ€œΞ©ξ€œπ’œπ‘‘||||β„Ž(π‘‘βˆ’π‘ )βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠)2+1𝑑𝑠𝑑π‘₯2ξ€·1βˆ’β„Žβˆ—ξ€Έξ€œπ’¬π‘‘β€–β„Ž(π‘‘βˆ’π‘ )β€–βˆ‡π‘’(𝑠)22πΆπ‘‘π‘ βˆ’π‘4𝛿3[β„Ž]ξ€œπ΅π‘‰Ξ©ξ€·β„Žξ…žξ€Έ+ξ€·β–‘βˆ‡π‘’π‘‘π‘₯1+𝛿2ξ€Έξ‚΅ξ€œπ’¬π‘‘ξ‚Άξ€œβ„Ž(π‘‘βˆ’π‘ )π‘‘π‘ Ξ©ξ€œπ’¬π‘‘||||β„Ž(π‘‘βˆ’π‘ )βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠)2+2𝑑𝑠𝑑π‘₯𝑝+1𝛿4𝐢𝑒𝑏2(𝑑)(1βˆ’πœ…)𝑝+1ℰ𝑝+1𝐢(𝑑)+π‘πœ…4𝛿4ξ€œΞ©(β„Žβ–‘βˆ‡π‘’)𝑑π‘₯+𝛿5π‘Ž(𝑑)β€–βˆ‡π‘’β€–22(π‘ž+1)+2π‘žβˆ’1πœ…π‘Ž(𝑑)𝛿5(1βˆ’πœ…)π‘žβ„°π‘ž+1+𝐢(𝑑)𝑝4𝛿3ξ‚΅ξ€œπ’¬π‘‘πœ‰ξ‚Άξ€œ(π‘‘βˆ’π‘ )π‘‘π‘ Ξ©ξ€œπ’¬π‘‘πœ‰||||(π‘‘βˆ’π‘ )βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠)2𝑑𝑠𝑑π‘₯.(3.16)

In the following theorem we will assume that 𝑝<π‘ž just to fix ideas. The result is also valid for 𝑝>π‘ž. It suffices to interchange π‘β†”π‘ž and 𝐴(𝑑)↔𝐡(𝑑) in the proof following it. The case 𝑝=π‘ž is easier.

We will make use of the following hypotheses for some positive constants 𝐴,𝐡,π‘ˆ, and 𝑉 to be determined.(A)π‘Ž(𝑑) is a continuously differentiable function such that π‘Žξ…ž(𝑑)<π΄π‘Ž(𝑑),𝑑β‰₯0.(B)𝑏(𝑑) is a continuously differentiable function such that π‘ξ…ž(𝑑)<𝐡𝑏(𝑑),𝑑β‰₯0.(C)𝑝>0 if 𝑛=1,2 and 0<𝑝<2/(π‘›βˆ’2) if 𝑛β‰₯3.(D)[∫∞0π‘Ž(𝑠)π‘’βˆ’π‘žπ‘ π‘‘π‘ ]1/π‘ž[∫∞0𝑏2(𝑠)π‘’βˆ’π‘π‘ π‘‘π‘ ]1/𝑝<π‘ˆ. (E)[∫∞0π‘Ž(𝑠)π‘’βˆ«βˆ’π‘žπ‘ 0πœ‚(𝜏)π‘‘πœπ‘‘π‘ ]1/π‘ž[∫∞0𝑏2(𝑠)π‘’βˆ«βˆ’π‘π‘ 0πœ‚(𝜏)π‘‘πœπ‘‘π‘ ]1/𝑝<𝑉.

Theorem 3.2. Assume that the hypotheses (H1)–(H3), (A)–(C) hold and β„›β„Ž<1/4. If limπ‘‘β†’βˆžπœ‚(𝑑)=πœ‚β‰ 0, then, for global solutions and small βˆ«π’¬πœ‰(𝑠)𝑑𝑠, there exist 𝑇1>0 and π‘ˆ>0 such that 𝐿(𝑑)>π‘ˆ,𝑑β‰₯𝑇1 or 𝐸(𝑑)≀𝑀1π‘’βˆ’πœˆ1𝑑,𝑑β‰₯0(3.17) for some positive constants 𝑀1 and 𝜈1 as long as (D) holds. If πœ‚=0, then there exist 𝑇2>0 and 𝑉>0 such that 𝐿(𝑑)>𝑉,𝑑β‰₯𝑇2 or 𝐸(𝑑)≀𝑀2π‘’βˆ’πœˆ2βˆ«π‘‘0πœ‚(𝑠)𝑑𝑠,𝑑β‰₯0(3.18) for some positive constants 𝑀2 and 𝜈2 as long as (E) holds.

Proof. A differentiation of Ξ¦1(𝑑) with respect to 𝑑 along trajectories of (1.1) gives Ξ¦ξ…ž1(‖‖𝑒𝑑)∢=𝑑‖‖22βˆ’β€–βˆ‡π‘’β€–22+ξ€œΞ©ξ€œβˆ‡π‘’β‹…π‘‘0β„Ž(π‘‘βˆ’π‘ )βˆ‡π‘’(𝑠)𝑑𝑠𝑑π‘₯βˆ’π‘Ž(𝑑)β€–βˆ‡π‘’β€–22(π‘ž+1)βˆ’π‘(𝑑)‖𝑒‖𝑝+2𝑝+2,(3.19) and Lemma 2.2 implies Ξ¦ξ…ž1(‖‖𝑒𝑑)≀𝑑‖‖22βˆ’ξ‚€πœ…1βˆ’2ξ‚β€–βˆ‡π‘’β€–22+12ξ€œπ‘‘0β„Ž(π‘‘βˆ’π‘ )β€–βˆ‡π‘’(𝑠)β€–22βˆ’1𝑑𝑠2ξ€œΞ©(β„Žβ–‘βˆ‡π‘’)𝑑π‘₯βˆ’π‘Ž(𝑑)β€–βˆ‡π‘’β€–22(π‘ž+1)βˆ’π‘(𝑑)‖𝑒‖𝑝+2𝑝+2,𝑑β‰₯0.(3.20) Next, a differentiation of Ξ¦3(𝑑) and Ξ¦4(𝑑) yields Ξ¦ξ…ž3(𝑑)=𝐻𝛾(0)β€–βˆ‡π‘’β€–22+ξ€œπ‘‘0π»ξ…žπ›Ύ(π‘‘βˆ’π‘ )β€–βˆ‡π‘’(𝑠)β€–22𝑑𝑠=𝐻𝛾(0)β€–βˆ‡π‘’β€–22βˆ’ξ€œπ‘‘0π›Ύξ…ž(π‘‘βˆ’π‘ )𝐻𝛾(π‘‘βˆ’π‘ )𝛾(π‘‘βˆ’π‘ )β€–βˆ‡π‘’(𝑠)β€–22ξ€œπ‘‘π‘ βˆ’π‘‘0β„Ž(π‘‘βˆ’π‘ )β€–βˆ‡π‘’(𝑠)β€–22𝑑𝑠≀𝐻𝛾(0)β€–βˆ‡π‘’β€–22βˆ’πœ‚(𝑑)Ξ¦3ξ€œ(𝑑)βˆ’π‘‘0β„Ž(π‘‘βˆ’π‘ )β€–βˆ‡π‘’(𝑠)β€–22Φ𝑑𝑠,𝑑β‰₯0,ξ…ž4(𝑑)=Ψ𝛾(0)β€–βˆ‡π‘’β€–22+ξ€œπ‘‘0Ψ′𝛾(π‘‘βˆ’π‘ )β€–βˆ‡π‘’(𝑠)β€–22𝑑𝑠=Ψ𝛾(0)β€–βˆ‡π‘’β€–22βˆ’ξ€œπ‘‘0π›Ύξ…ž(π‘‘βˆ’π‘ )Ψ𝛾(π‘‘βˆ’π‘ )𝛾(π‘‘βˆ’π‘ )β€–βˆ‡π‘’(𝑠)β€–22ξ€œπ‘‘π‘ βˆ’π‘‘0πœ‰(π‘‘βˆ’π‘ )β€–βˆ‡π‘’(𝑠)β€–22𝑑𝑠≀Ψ𝛾(0)β€–βˆ‡π‘’β€–22βˆ’πœ‚(𝑑)Ξ¦4ξ€œ(𝑑)βˆ’π‘‘0πœ‰(π‘‘βˆ’π‘ )β€–βˆ‡π‘’(𝑠)β€–22𝑑𝑠,𝑑β‰₯0.(3.21)
Taking into account Lemma 3.1 and the relations (2.6), (3.20)-(3.21), we see that πΏξ…ž(ξ‚΅1𝑑)≀2βˆ’πΆπ‘4𝛿3πœ†2[β„Ž]ξ‚Άξ€œπ΅π‘‰Ξ©ξ€·β„Žξ…žξ€Έξ€Ίπœ†β–‘βˆ‡π‘’π‘‘π‘₯+1+𝛿3βˆ’β„Žβˆ—ξ€Έπœ†2‖‖𝑒𝑑‖‖22+ξ‚»πœ†2ξ€·1βˆ’β„Žβˆ—ξ€Έξ‚Έπ›Ώ1+32ξ€œπ’¬π‘‘ξ‚Ήβ„Ž(π‘‘βˆ’π‘ )𝑑𝑠+πœ†3𝐻𝛾(0)+πœ†4Ψ𝛾(0)βˆ’πœ†1ξ‚€πœ…1βˆ’2ξ‚ξ‚ΌΓ—β€–βˆ‡π‘’β€–22+ξƒ©πœ†12+πœ†2ξ€·1βˆ’β„Žβˆ—ξ€Έ2βˆ’πœ†3ξƒͺξ€œπ‘‘0β„Ž(π‘‘βˆ’π‘ )β€–βˆ‡π‘’(𝑠)β€–22+ξ‚Έπ‘Žπ‘‘π‘ ξ…ž(𝑑)2(π‘ž+1)+𝛿5πœ†2π‘Ž(𝑑)βˆ’πœ†1ξ‚Ήπ‘Ž(𝑑)β€–βˆ‡π‘’β€–22(π‘ž+1)+ξ‚Έξ€·1+𝛿2ξ€Έπœ†2ξ€œπ’¬π‘‘πœ†β„Ž(π‘‘βˆ’π‘ )π‘‘π‘ βˆ’12ξ‚ΉΓ—ξ€œΞ©(β„Žβ–‘βˆ‡π‘’)𝑑π‘₯+πœ†2πœ…ξ‚Έ1+1βˆ’β„Žβˆ—4𝛿1+1𝛿2ξ‚Ήξ€œΞ©ξ€œπ’œπ‘‘||||β„Ž(π‘‘βˆ’π‘ )βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠)2𝑑𝑠𝑑π‘₯βˆ’πœ†3πœ‚(𝑑)Ξ¦3(𝑑)βˆ’πœ†4πœ‚(𝑑)Ξ¦4𝑏(𝑑)+ξ…ž(𝑑)𝑝+2βˆ’πœ†1𝑏(𝑑)‖𝑒‖𝑝+2𝑝+2+πœ†2πΆπ‘πœ…4𝛿4ξ€œΞ©+2(β„Žβ–‘βˆ‡π‘’)𝑑π‘₯π‘žβˆ’1πœ…π‘Ž(𝑑)𝛿5(1βˆ’πœ…)π‘žπœ†2β„°π‘ž+12(𝑑)+𝑝+1𝛿4𝐢𝑒𝑏2(𝑑)(1βˆ’πœ…)𝑝+1πœ†2ℰ𝑝+1(𝑑)βˆ’πœ†4ξ€œπ‘‘0πœ‰(π‘‘βˆ’π‘ )β€–βˆ‡π‘’(𝑠)β€–22+πœ†π‘‘π‘ 2𝐢𝑝4𝛿3ξ‚΅ξ€œπ’¬π‘‘ξ‚Άξ€œπœ‰(π‘‘βˆ’π‘ )π‘‘π‘ π’¬π‘‘πœ‰(π‘‘βˆ’π‘ )β€–βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠)β€–22𝑑𝑠,𝑑β‰₯π‘‘βˆ—.(3.22) Next, as in [17], we introduce the sets π’œπ‘›ξ€½βˆΆ=π‘ βˆˆπ‘+βˆΆπ‘›β„Žξ…žξ€Ύ(𝑠)+β„Ž(𝑠)≀0,π‘›βˆˆπ,(3.23) and observe that ξšπ‘›π’œπ‘›=𝐑+β§΅ξ€½π’¬β„Žβˆͺπ’©β„Žξ€Ύ,(3.24) where π’©β„Ž is the null set where β„Žξ…ž is not defined and π’¬β„Ž is as in (3.2). Furthermore, if we denote π’¬π‘›βˆΆ=𝐑+β§΅π’œπ‘›, then limπ‘›β†’βˆžξβ„Ž(𝒬𝑛)=β„Ž(π’¬β„Ž) because 𝒬𝑛+1βŠ‚π’¬π‘› for all 𝑛 and ⋂𝑛𝒬𝑛=π’¬β„Žβˆͺπ’©β„Ž. Moreover, we designate by 𝐴𝑛𝑑 the sets ξ‚π΄π‘›π‘‘ξ€½βˆΆ=π‘ βˆˆπ‘+∢0≀𝑠≀𝑑,π‘›β„Žξ…ž(ξ€Ύπ‘‘βˆ’π‘ )+β„Ž(π‘‘βˆ’π‘ )≀0,π‘›βˆˆπ.(3.25) In (3.22), we take π’œπ‘‘ξ‚π΄βˆΆ=𝑛𝑑 and π’¬π‘‘ξ‚π’¬βˆΆ=𝑛𝑑. Choosing πœ†1=(β„Žβˆ—βˆ’πœ€)πœ†2, it is clear that ξ€·1+𝛿2ξ€Έπœ†2πœ…ξβ„Žξ€·π’¬π‘›ξ€Έβˆ’πœ†12≀0(3.26) for small πœ€ and 𝛿2, large 𝑛 and π‘‘βˆ—, if ξβ„Ž(𝒬)<1/4. We deduce that ξ€·1+𝛿2ξ€Έπœ†2ξ€œξ‚π’¬π‘›π‘‘πœ†β„Ž(π‘‘βˆ’π‘ )π‘‘π‘ βˆ’12<0.(3.27) Furthermore, if ξβ„Ž(𝒬)<1/4, then 3ξ€·1βˆ’β„Žβˆ—ξ€Έ2ξ€œξ‚π’¬π‘›π‘‘β„Ž(π‘‘βˆ’π‘ )𝑑𝑠<π›Ώβ„Žβˆ—ξ‚€πœ…1βˆ’2(3.28) with 3𝛿=1βˆ’β„Žβˆ—ξ€Έπœ…4(2βˆ’πœ…)β„Žβˆ—+𝛽(3.29) and a small 𝛽>0. Pick πœ†3=12ξ€Ίπœ†1+πœ†2ξ€·1βˆ’β„Žβˆ—ξ€Έξ€»(3.30) and 𝐻𝛾(0) such that πœ†3𝐻𝛾(0)<πœ†2(1βˆ’π›Ώ)β„Žβˆ—(2βˆ’πœ…)2.(3.31) Note that this is possible if π‘‘βˆ— is so large that β„Žβˆ—>7πœ…/(8βˆ’πœ…) even though 𝐻𝛾(0)=𝛾(0)βˆ’1ξ€œβˆž0ξ€œβ„Ž(𝑠)𝛾(𝑠)𝑑𝑠β‰₯∞0β„Ž(𝑠)𝑑𝑠=πœ….(3.32)
Taking the relations (3.22)–(3.30) into account and selecting πœ†2<𝛿3/𝐢𝑝𝐡𝑉[β„Ž] so that 12βˆ’πΆπ‘πœ†24𝛿3[β„Ž]β‰₯1𝐡𝑉4,(3.33) and small enough so that πœ†2πœ…ξ‚΅1+1βˆ’β„Žβˆ—4𝛿1+1𝛿2+𝐢𝑝4𝛿4ξ‚Ά<1,4𝑛(3.34) we find for 𝛿3=πœ€/2, large 𝛿4, small Ψ𝛾(0), and 𝑑β‰₯π‘‘βˆ—πΏξ…ž(𝑑)β‰€βˆ’πΆ1‖‖𝑒𝑑‖‖22+β€–βˆ‡π‘’β€–22+ξ€œΞ©ξ‚Ό+2(β„Žβ–‘βˆ‡π‘’)𝑑π‘₯𝑝+1𝛿4𝐢𝑒𝑏2(𝑑)(1βˆ’πœ…)𝑝+1πœ†2ℰ𝑝+1+ξ‚Έπ‘Ž(𝑑)ξ…ž(𝑑)2(π‘ž+1)+𝛿5πœ†2π‘Ž(𝑑)βˆ’πœ†1ξ‚Ήπ‘Ž(𝑑)β€–βˆ‡π‘’β€–22(π‘ž+1)+2π‘žβˆ’1πœ…π‘Ž(𝑑)𝛿5(1βˆ’πœ…)π‘žπœ†2β„°π‘ž+1(𝑑)βˆ’πœ†3πœ‚(𝑑)Ξ¦3(𝑑)βˆ’πœ†4πœ‚(𝑑)Ξ¦4(1𝑑)+2ξ‚Έπœ†1+2𝐢𝑝2𝛿3ξ€œπ’¬π‘‘ξ‚Ήξ€œπœ‰(π‘‘βˆ’π‘ )π‘‘π‘ Ξ©ξ€œπ’¬π‘›π‘‘β„Žξ…ž(||||π‘‘βˆ’π‘ )βˆ‡π‘’(𝑑)βˆ’βˆ‡π‘’(𝑠)2+𝑏𝑑𝑠𝑑π‘₯ξ…ž(𝑑)𝑝+2βˆ’πœ†1‖𝑏(𝑑)𝑒‖𝑝+2𝑝+2βˆ’πœ†4ξ€œπ‘‘0πœ‰(π‘‘βˆ’π‘ )β€–βˆ‡π‘’(𝑠)β€–22𝑑𝑠(3.35) for some positive constant 𝐢1. Take πœ†4>1+(πœ†2𝐢𝑝/2𝛿3)βˆ«π’¬πœ‰(𝑠)𝑑𝑠,𝛿5,βˆ«π’¬πœ‰(𝑠)𝑑𝑠 small, and π‘Žξ…ž(𝑑)<ξ€·πœ†2(π‘ž+1)1ξ€Έβˆ’π›Όπ‘Ž(𝑑)(3.36) (i.e., 𝐴=2(π‘ž+1)(πœ†1βˆ’π›Ό) for some 0<𝛼<πœ†1) and π‘ξ…ž(𝑑)<ξ€·πœ†π‘+21ξ€Έπ‘βˆ’π›½(𝑑)(3.37) (i.e., 𝐡=(𝑝+2)(πœ†1βˆ’π›½) for some 0<𝛽<πœ†1) to derive that πΏξ…ž(𝑑)β‰€βˆ’πΆ22β„°(𝑑)+𝑝+1𝛿4𝐢𝑒𝑏2(𝑑)(1βˆ’πœ…)𝑝+1πœ†2ℰ𝑝+12(𝑑)+π‘žβˆ’1πœ…π‘Ž(𝑑)𝛿5(1βˆ’πœ…)π‘žπœ†2β„°π‘ž+1(𝑑)βˆ’πœ†3πœ‚(𝑑)Ξ¦3(𝑑)βˆ’πœ†4πœ‚(𝑑)Ξ¦4(𝑑)(3.38) for some positive constant 𝐢2.
If limπ‘‘β†’βˆžπœ‚(𝑑)β‰ 0, then there exist a ̂𝑑β‰₯π‘‘βˆ— and 𝐢3>0 such that πœ‚(𝑑)β‰₯𝐢3 for ̂𝑑𝑑β‰₯. Thus, in virtue of Proposition 2.1, for 𝐢3>0, we have πΏξ…ž(𝑑)β‰€βˆ’πΆ3𝐿(𝑑)+𝐡(𝑑)𝐿𝑝+1(𝑑)+𝐴(𝑑)πΏπ‘ž+1(𝑑),(3.39) where 2𝐡(𝑑)∢=𝑝+1𝛿4πΆπ‘’πœ†2(1βˆ’πœ…)𝑝+1𝜌1𝑝+1𝑏22(𝑑),𝐴(𝑑)∢=π‘žβˆ’1πœ…πœ†2𝛿5(1βˆ’πœ…)π‘žπœŒ1π‘ž+1π‘Ž(𝑑).(3.40) If there exists a ̂𝑑𝑇β‰₯ such that ∫𝐿(𝑇)<[π‘βˆž0𝐡(𝑠)𝑑𝑠]βˆ’1/𝑝, where 𝐡(𝑑)∢=𝐡(𝑑)π‘’βˆ’π‘πΆ3(π‘‘βˆ’π‘‡), then from (3.39) 𝑒𝐢3(π‘‘βˆ’π‘‡)𝐿(𝑑)ξ…žβ‰€π‘’πΆ3(π‘‘βˆ’π‘‡)𝐡(𝑑)𝐿𝑝+1(𝑑)+𝑒𝐢3(π‘‘βˆ’π‘‡)𝐴(𝑑)πΏπ‘ž+1(𝑑),(3.41) and it follows that for 𝑑β‰₯π‘‡ξ‚ξ€œπΏ(𝑑)≀𝐿(𝑇)+π‘‘π‘‡π‘’βˆ’π‘πΆ3(π‘ βˆ’π‘‡)𝐿𝐡(𝑠)𝑝+1(ξ€œπ‘ )𝑑𝑠+π‘‘π‘‡π‘’βˆ’π‘žπΆ3(π‘ βˆ’π‘‡)𝐿𝐴(𝑠)π‘ž+1(𝑠)𝑑𝑠(3.42) with 𝐿(𝑑)∢=𝑒𝐢3(π‘‘βˆ’π‘‡)𝐿(𝑑). Now we apply Lemma 2.3 to get 𝐿𝑁(𝑑)β‰€βˆ’π‘žξ€œβˆ’π‘žπ‘‘0𝐴(𝑠)π‘‘π‘ βˆ’1/π‘ž,𝑑β‰₯𝑇,(3.43) where 𝐴(𝑑)∢=π‘’βˆ’π‘žπΆ3(π‘‘βˆ’π‘‡)𝐴(𝑑) and π‘βˆΆ=[𝐿(𝑇)βˆ’π‘βˆ«βˆ’π‘βˆž0𝐡(𝑠)𝑑𝑠]βˆ’1/𝑝. If, in addition, π‘žβˆ«βˆž0𝐴(𝑠)𝑑𝑠<𝐿(𝑇)βˆ’π‘ž, then 𝐿(𝑑) is uniformly bounded by a positive constant 𝐢4. Thus 𝐿(𝑑)≀𝐢4π‘’βˆ’πΆ3𝑑,𝑑β‰₯𝑇,(3.44) and by continuity (3.44) holds for all 𝑑β‰₯0.
If limπ‘‘β†’βˆžπœ‚(𝑑)=0, then for any 𝐢>0 there exists a 𝑑(𝐢)β‰₯π‘‘βˆ— such that πœ‚(𝑑)≀𝐢 for 𝑑β‰₯𝑑(𝐢). Therefore, πΏξ…ž(𝑑)β‰€βˆ’πΆ5πœ‚(𝑑)𝐿(𝑑)+𝐡(𝑑)𝐿𝑝+1(𝑑)+𝐴(𝑑)πΏπ‘ž+1(𝑑),𝑑β‰₯𝑑=𝑑𝐢2ξ€Έ(3.45) for some 𝐢5>0. The previous argument carries out with 𝑒𝐢3(π‘‘βˆ’π‘‡) replaced by 𝑒𝐢5βˆ«π‘‘π‘‡πœ‚(𝑑)𝑑𝑠.
In case that π‘ž<𝑝, we reverse the roles of 𝑝 and π‘ž in the argument above. The case 𝑝=π‘ž is clear.

Remark 3.3. The case where the derivative of the kernel does not approach zero on π’œ (as is the case, for instance, when β„Žξ…žβ‰€βˆ’πΆβ„Ž on π’œ) is interesting. Indeed, the right-hand side in condition (3.34) will be replaced by 𝐢/4 with a possibly large constant 𝐢.

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 β€œsmall” subintervals and open the door for (optimal) estimations and improvements of these sets.

Remark 3.5. The assumptions π‘Žξ…ž(𝑑)<2(π‘ž+1)(πœ†1βˆ’π›Ό)π‘Ž(𝑑) and π‘ξ…ž(𝑑)<(𝑝+2)(πœ†1βˆ’π›½)𝑏(𝑑) may be relaxed to π‘Žξ…ž(𝑑)<2(π‘ž+1)πœ†1π‘Ž(𝑑) and π‘ξ…ž(𝑑)<(𝑝+2)πœ†1𝑏(𝑑), respectively. In this case 𝛼=𝛼(𝑑) and 𝛽=𝛽(𝑑) would depend on 𝑑.

Remark 3.6. The assertion in Theorem 3.2 is an β€œalternative” 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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