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Hindawi Publishing Corporation Journal of Applied Mathematics Volume 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 Correspondence should be addressed to Nasser-Eddine Tatar, [email protected] Received 3 June 2012; Accepted 17 July 2012 Academic Editor: Yongkun Li Copyright q 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. 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 m ⎛ bi t|u|pi u ⎝1 i1 k ⎞ 2qj aj t∇u2 ⎠Δu j1 − t ht − sΔusds, in Ω × R , 1.1 0 u 0, ux, 0 u0 x, on Γ × R , ut x, 0 u1 x, in Ω, 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 2 Journal of Applied Mathematics the nonnegative functions aj t, bi t, and ht 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 h. 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 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 ∈ H01 Ω × L2 Ω and ht is a nonnegative summable kernel. If 0 < p < 2/n − 2 when n ≥ 3 and p > 0 when n 1, 2, then there exists a unique solution u to problem 1.1 such that u ∈ C 0, T ; H01 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 Et at bt 1 2q1 p2 ∇u2 ut 22 ∇u22 up2 , 2 p2 2 q1 t ≥ 0, 2.1 Journal of Applied Mathematics 3 where · p denotes the norm in Lp Ω. Then by 1.1 it is easy to see that for t ≥ 0 E t Ω ∇ut · t a t b t 2q1 p2 ht − s∇usds dx ∇u2 up2 . p2 2 q1 0 2.2 The first term in the right-hand side of 2.2 may be written as the derivative of some expression; namely, Ω ∇ut · t ht − s∇usds dx 0 1 h ∇u dx − ht∇u22 2 Ω t 1 d 2 − hsds ∇u2 , h∇udx − 2 dt Ω 0 1 2 2.3 where hvt : t ht − s|vt − vs|2 ds. 2.4 0 Therefore, if we modify Et to t 1 2 2 Et : ut 2 1 − hsds ∇u2 h∇udx 2 0 Ω 2.5 bt at 2q1 p2 ∇u2 up2 , p2 2 q1 we obtain for t ≥ 0 1 E t 2 Ω a t b t 2q1 p2 h ∇u − ht|∇u|2 dx ∇u2 up2 . p2 2 q1 2.6 Assuming that 1− ∞ hsds : 1 − κ > 0 2.7 0 makes Et a nonnegative functional. The following functionals Φ1 t : t Φ2 t : − Ω ut 0 Ω ut u dx, ht − sut − usds dx 2.8 4 Journal of Applied Mathematics are standard and will be used here. The next ones have been introduced by the present author in 24 Φ3 t : t Hγ t − 0 s∇us22 ds, Φ4 t : t 0 Ψγ t − s∇us22 ds, 2.9 where Hγ t : γt −1 ∞ hsγsds, −1 Ψγ t : γt t ∞ ξsγsds, t ≥ 0, 2.10 t and γt and ξt are two nonnegative functions which will be precised later see H2, H3. The functional Lt : Et 4 λi Φi t 2.11 i1 for some λi > 0, i 1, 2, 3, 4, to be determined is equivalent to Et Φ3 t Φ4 t. Proposition 2.1. There exist ρi > 0, i 1, 2 such that ρ1 Et Φ3 t Φ4 t ≤ Lt ≤ ρ2 Et Φ3 t Φ4 t 2.12 for all t ≥ 0 and small λi , i 1, 2. Proof. By the inequalities Cp 1 ut 22 ∇u22 , 2 2 Ω Cp κ 1 2 Φ2 t ≤ ut 2 h∇udx, 2 2 Ω Φ1 t ut u dx ≤ 2.13 where Cp is the Poincaré constant, we have t 1 bt 1 p2 2 1 − hsds λ1 Cp ∇u22 Lt ≤ 1 λ1 λ2 ut 2 up2 2 2 p 2 0 1 at 2q1 1 λ2 Cp κ λ3 Φ3 t λ4 Φ4 t, ∇u2 h∇udx 2 2 q1 Ω t ≥ 0. 2.14 Journal of Applied Mathematics 5 On the other hand, 2Lt ≥ 1 − λ1 − λ2 ut 22 1 − λ2 Cp κ Ω 2bt p2 up2 p2 h∇udx at 2q1 1 − κ − λ1 Cp ∇u22 2λ3 Φ3 t 2λ4 Φ4 t. ∇u2 q1 2.15 Therefore, ρ1 Et Φ3 t Φ4 t ≤ Lt ≤ ρ2 Et Φ3 t Φ4 t for some constant ρi > 0, i 1, 2 and small λi , i 1, 2 such that λ1 < min{1, 1 − κ/Cp } and λ2 < min{1/Cp κ, 1 − λ1 }. The identity to follow is easy to justify and is helpful to prove our result. Lemma 2.2. One has for h ∈ C0, ∞ and v ∈ C0, ∞; L2 Ω t 1 vt ht − svsds dx 2 0 Ω t 0 1 2 hsds vt22 t 1 ht − svs22 ds − 2 0 2.16 Ω hvdx, t ≥ 0. The next lemma is crucial in estimating partially our nonlinear terms. It can be found in 47. Let I ⊂ R, and let g1 , g2 : I → R \ {0}. We write g1 ∝ g2 if g2 /g1 is nondecreasing in I. Lemma 2.3. Let at 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 ut is a nonnegative continuous functions in J. If g1 ∝ g2 ∝ · · · ∝ gn in 0, ∞, then the inequality ut ≤ at n t j1 kj sgj usds, t ∈ J, α 2.17 implies that ut ≤ cn t, 2.18 α ≤ t < β0 , where c0 t : sup0≤s≤t as, cj t : G−1 j Gj u : Gj cj−1 t t kj sds , j 1, . . . , n, α u uj dx , gj x u > 0 uj > 0, j 1, ..., n , and β0 is chosen so that the functions cj t, j 1, . . . , n are defined for α ≤ t < β0 . 2.19 6 Journal of Applied Mathematics Lemma 2.4. Assume that 2 ≤ q < ∞ if n 1, 2 or 2 ≤ q < 2n/n − 2 if n ≥ 3. Then there exists a positive constant Ce Ce Ω, q such that uq ≤ Ce ∇u2 2.20 for u ∈ H01 Ω. 3. Asymptotic Behavior In this section we state and prove our result. To this end we need some notation. For every by measurable set A ⊂ R , we define the probability measure h 1 hA : κ A hsds. 3.1 The nondecreasingness set and the non-decreasingness rate of h are defined by Qh : s ∈ R : hs > 0, h s ≥ 0 , 3.2 h , Rh : hQ 3.3 respectively. The following assumptions on the kernel ht will be adopted. H1 ht ≥ 0 for all t ≥ 0 and 0 < κ ∞ 0 hsds < 1. H2 h is absolutely continuous and of bounded variation on 0, ∞ and h t ≤ ξt for some nonnegative summable function ξt max{0, h t} where h t exists and almost all t > 0. H3 There exists a nondecreasing that γ t/γt ηt is a ∞ function γt > 0 such ∞ nonincreasing function: 0 hsγsds < ∞ and 0 ξsγsds < ∞. 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. t Let t∗ > 0 be a number such that 0∗ hsds h∗ > 0. We denote by Bt the set Bt : B ∩ 0, t. Lemma 3.1. One has for t ≥ t∗ and δi > 0, i 1, . . . , 5 Φ2 t 3 ht − sds ∇u22 δ3 − h∗ ut 22 ≤1 − h∗ δ1 2 Qt κ 1 1 − h∗ 1 ht − s|∇ut − ∇us|2 ds dx κ 4δ1 δ2 Ω At Journal of Applied Mathematics Cp 1 1 − h∗ ht − s∇us22 ds − BV h h ∇u dx 2 4δ3 Qt Ω 1 δ2 Qt ht − sds 2p1 δ4 Ce b2 t p1 Ω ht − s|∇ut − ∇us|2 ds dx Cp κ 4δ4 Ep1 t Qt 7 2q1 h∇udx δ5 at∇u2 1 − κ Ω Cp ξt − sds ξt − s|∇ut − ∇us|2 ds dx, 4δ3 Qt Ω Qt 2q−1 κat q1 E t δ5 1 − κq 3.4 where BV h is the total variation of h. 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 t − Ω utt t − Ω ht − sut − usds dx 0 ut h t − sut − usds ut 0 3.5 t hsds dx 0 or Φ2 t − 1− Ω t 0 t 2q hsds Δu − bt|u|p u at∇u2 Δu t ht − sΔut − Δusds 0 − t 0 ht − sut − usds dx 0 t hsds ut 22 − Ω ut h t − sut − usds dx, 0 Therefore, Φ2 t 1− t hsds 0 × Ω ∇u · t 0 2q at∇u2 ht − s∇ut − ∇usds dx − t 0 hsds ut 22 t ≥ 0. 3.6 8 Journal of Applied Mathematics bt Ω |u|p u t t ht − sut − usds dx − 0 2 t ht − s∇ut − ∇usds dx, 0 Ω Ω ut h t − sut − usds dx 0 t ≥ 0. 3.7 For all measurable sets A and Q such that A R \ Q, it is clear that Ω ∇u · t ht − s∇ut − ∇usds dx 0 Ω ∇u · Ω ht − s∇ut − ∇usds dx Ω A∩0,t ∇u · ∇u · Q∩0,t A∩0,t Q∩0,t ht − s∇ut − ∇usds dx 3.8 ht − s∇ut − ∇usds dx ht − sds ∇u22 − Ω ∇u · Q∩0,t ht − s∇usds dx, t ≥ 0. For δ1 > 0, the first term in the right-hand side of 3.8 satisfies Ω ∇u · ≤ At ht − s∇ut − ∇usds dx δ1 ∇u22 κ 4δ1 3.9 2 Ω At ht − s|∇ut − ∇us| ds dx, t ≥ 0, and the third one fulfills Ω ∇u · 1 ≤ 2 Qt ht − s∇usds dx Qt ht − sds ∇u22 1 2 3.10 Qt ht − s∇us22 ds, t ≥ 0. Journal of Applied Mathematics 9 Back to 3.8 we may write Ω ∇u · t ht − s∇ut − ∇usds dx 0 κ ht − s|∇ut − ∇us|2 ds dx 4δ1 Ω At 3 1 2 ∇u2 ht − sds ht − s∇us22 ds, 2 2 Qt Qt ≤ δ1 ∇u22 3.11 t ≥ 0. The last term in the right-hand side of 3.7 will be estimated as follows: 2 t ht − s∇ut − ∇usds dx Ω 0 1 ht − s|∇ut − ∇us|2 ds dx κ ≤ 1 δ2 Ω At 1 δ2 Qt ht − sds Ω Qt 3.12 ht − s|∇ut − ∇us|2 ds dx, δ2 > 0. For the fourth term in 3.7, it holds that − t Ω ut h t − sut − usds dx 0 ≤ δ3 ut 22 − Cp 4δ3 Cp BV h 4δ3 Qt h ∇u dx Ω 3.13 ξt − sds ξt − s|∇ut − ∇us|2 ds dx, Ω Qt δ3 > 0, t ≥ 0. Moreover, from Lemma 2.4, for p > 0 if n 1, 2 and 0 < p < 2/n − 2 if n ≥ 3, we find bt Ω |u|p u ≤ ht − sut − usds dx 0 ≤ δ4 b ≤ t 2 2p1 tu2p1 Cp 4δ4 2p1 δ4 Ce b2 t∇u2 2p1 δ4 Ce b2 t 1 − κp1 t hsds 0 Cp κ 4δ4 Ep1 t Ω h∇udx 3.14 Cp κ 4δ4 Ω h∇udx Ω h∇udx, δ4 > 0, t ≥ 0. 10 Journal of Applied Mathematics The definition of Et in 2.5 allows us to write 2q at∇u2 ≤ ∇u · Ω t ht − s∇ut − ∇usds dx 0 2q at∇u2 κ 2 δ5 ∇u2 h∇udx 4δ5 Ω 2q1 ≤ δ5 at∇u2 2q−1 κat q1 E t, δ5 1 − κq 3.15 δ5 > 0, t ≥ 0. Gathering all the relations 3.11–3.15 together with 3.7, we obtain for t ≥ t∗ Φ2 t 3 ht − sds ∇u22 δ3 − h∗ ut 22 ≤ 1 − h∗ δ1 2 Qt κ 1 κ 1 − h∗ 1 ht − s|∇ut − ∇us|2 ds dx 4δ1 δ2 Ω At Cp 1 1 − h∗ ht − s∇us22 ds − BV h h ∇u dx 2 4δ3 Qt Ω 1 δ2 Qt ht − sds 2p1 δ4 Ce b2 t p1 Ep1 t Ω Cp κ 4δ4 Qt ht − s|∇ut − ∇us|2 ds dx 2q1 h∇udx δ5 at∇u2 1 − κ Ω Cp ξt − sds ξt − s|∇ut − ∇us|2 ds dx. 4δ3 Qt Ω Qt 2q−1 κat q1 E t δ5 1 − κq 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 p ↔ q and At ↔ Bt 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 at is a continuously diﬀerentiable function such that a t < Aat, t ≥ 0. B bt is a continuously diﬀerentiable function such that b t < Bbt, t ≥ 0. C p > 0 if n 1, 2 and 0 < p < 2/n − 2 if n ≥ 3. D E ∞ 0 ∞ 0 ase−qs ds ase−q s 0 1/q ητdτ ∞ 0 ds 1/p b2 se−ps ds 1/q ∞ 0 b2 se−p < U. s 0 ητdτ ds 1/p < V. Journal of Applied Mathematics 11 Theorem 3.2. Assume that the hypotheses(H1)–(H3), (A)–(C) hold and Rh < 1/4. If limt → ∞ ηt η/ 0, then, for global solutions and small Q ξsds, there exist T1 > 0 and U > 0 such that Lt > U, t ≥ T1 or Et ≤ M1 e−ν1 t , t≥0 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 Lt > V, t ≥ T2 or Et ≤ M2 e−ν2 t 0 ηsds 3.18 t≥0 , 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 t : ut 22 − − ∇u22 2q1 at∇u2 Ω − ∇u · t ht − s∇usds dx 0 3.19 p2 btup2 , and Lemma 2.2 implies κ 1 t Φ1 t ≤ ut 22 − 1 − ht − s∇us22 ds ∇u22 2 2 0 1 2q1 p2 − btup2 , − h∇udx − at∇u2 2 Ω 3.20 t ≥ 0. Next, a diﬀerentiation of Φ3 t and Φ4 t yields Φ3 t Hγ 0∇u22 t 0 Hγ t − s∇us22 ds t γ t − s Hγ t − s∇us22 ds − γt − s 0 t ≤ Hγ 0∇u22 − ηtΦ3 t − ht − s∇us22 ds, Hγ 0∇u22 − Φ4 t Ψγ 0∇u22 t 0 0 ht − s∇us22 ds t ≥ 0, 0 Ψγ t − 3.21 s∇us22 ds t γ t − s Ψγ t − s∇us22 ds − 0 γt − s t 2 ≤ Ψγ 0∇u2 − ηtΦ4 t − ξt − s∇us22 ds, Ψγ 0∇u22 − t 0 t 0 ξt − s∇us22 ds t ≥ 0. 12 Journal of Applied Mathematics Taking into account Lemma 3.1 and the relations 2.6, 3.20-3.21, we see that L t ≤ 1 Cp − λ2 BV h 2 4δ3 3 λ2 1 − h∗ δ1 2 × ∇u22 h ∇u dx λ1 δ3 − h∗ λ2 ut 22 Ω κ ht − sds λ3 Hγ 0 λ4 Ψγ 0 − λ1 1 − 2 Qt t λ1 λ2 1 − h∗ − λ3 ht − s∇us22 ds 2 2 0 a t λ1 2q1 1 δ2 λ2 ht − sds − δ5 λ2 at − λ1 at ∇u2 2 2 q1 Qt 3.22 1 − h∗ 1 2 × h∇udx λ2 κ 1 ht − s|∇ut − ∇us| ds dx 4δ1 δ2 Ω At Ω λ2 Cp κ b t p2 − λ1 bt up2 − λ3 ηtΦ3 t − λ4 ηtΦ4 t h∇udx p2 4δ4 Ω t 2p1 δ4 Ce b2 t 2q−1 κat q1 p1 λ E λ E ξt − s∇us22 ds − λ t t 2 2 4 p1 δ5 1 − κq 1 − κ 0 λ2 Cp ξt − sds ξt − s∇ut − ∇us22 ds, t ≥ t∗ . 4δ3 Qt Qt Next, as in 17, we introduce the sets An : s ∈ R : nh s hs ≤ 0 , n ∈ N, 3.23 and observe that An R \ {Qh ∪ Nh }, 3.24 n where Nh is the null set where h is not defined and Qh is as in 3.2. Furthermore, if we denote h because Qn1 ⊂ Qn for all n and n hQ Qn : R \ An , then limn → ∞ hQ n Qn Qh ∪ Nh . ! Moreover, we designate by Ant the sets ! nt : s ∈ R : 0 ≤ s ≤ t, nh t − s ht − s ≤ 0 , A n ∈ N. 3.25 ! nt . Choosing λ1 h∗ − ελ2 , it is clear that ! nt and Qt : Q In 3.22, we take At : A n − 1 δ2 λ2 κhQ λ1 ≤0 2 3.26 Journal of Applied Mathematics 13 < 1/4. We deduce that for small ε and δ2 , large n and t∗ , if hQ 1 δ2 λ2 ! nt Q ht − sds − λ1 < 0. 2 3.27 Furthermore, if hQ < 1/4, then 31 − h∗ 2 κ ht − sds < δh∗ 1 − 2 ! nt Q 3.28 with 31 − h∗ κ β 42 − κh∗ 3.29 1 λ1 λ2 1 − h∗ 2 3.30 δ and a small β > 0. Pick λ3 and Hγ 0 such that λ3 Hγ 0 < λ2 1 − δh∗ 2 − κ . 2 3.31 Note that this is possible if t∗ is so large that h∗ > 7κ/8 − κ even though Hγ 0 γ0−1 ∞ 0 hsγsds ≥ ∞ hsds κ. 3.32 0 Taking the relations 3.22–3.30 into account and selecting λ2 < δ3 /Cp BV h so that 1 Cp λ2 1 − BV h ≥ , 2 4δ3 4 3.33 Cp 1 − h∗ 1 1 , < λ2 κ 1 4δ1 δ2 4δ4 4n 3.34 and small enough so that 14 Journal of Applied Mathematics we find for δ3 ε/2, large δ4 , small Ψγ 0, and t ≥ t∗ 2p1 δ4 Ce b2 t L t ≤ − C1 ut 22 ∇u22 h∇udx λ2 Ep1 t p1 1 − κ Ω 2q−1 κat a t 2q1 λ2 Eq1 t − λ3 ηtΦ3 t δ5 λ2 at − λ1 at ∇u2 δ5 1 − κq 2 q1 λ2 Cp 1 1 − λ4 ηtΦ4 t ξt − sds h t − s|∇ut − ∇us|2 ds dx 2 2δ3 Qt Ω Qnt t b t p2 − λ1 bt up2 − λ4 ξt − s∇us22 ds p2 0 for some positive constant C1 . Take λ4 > 1 λ2 Cp /2δ3 Q ξsds, δ5 , 3.35 Q ξsds small, and a t < λ1 − αat 2 q1 3.36 i.e., A 2q 1λ1 − α for some 0 < α < λ1 and b t < λ1 − β bt p2 3.37 i.e., B p 2λ1 − β for some 0 < β < λ1 to derive that L t ≤ − C2 Et 2p1 δ4 Ce b2 t 1 − κp1 λ2 Ep1 t 2q−1 κat λ2 Eq1 t δ5 1 − κq 3.38 − λ3 ηtΦ3 t − λ4 ηtΦ4 t for some positive constant C2 . 0, then there exist a t ≥ t∗ and C3 > 0 such that ηt ≥ C3 for t ≥ t. Thus, If limt → ∞ ηt / in virtue of Proposition 2.1, for C3 > 0, we have L t ≤ −C3 Lt BtLp1 t AtLq1 t, 3.39 where Bt : At : 2p1 δ4 Ce λ2 1 − p1 κp1 ρ1 b2 t, 2q−1 κλ2 q1 δ5 1 − κq ρ1 3.40 at. Journal of Applied Mathematics 15 If there exists a T ≥ t such that LT < p from 3.39 ∞ 0 −1/p ! ! , where Bt : Bte−pC3 t−T , then Bsds # " eC3 t−T Lt ≤ eC3 t−T BtLp1 t eC3 t−T AtLq1 t, 3.41 and it follows that for t ≥ T ! Lt ≤ LT t ! p1 sds e−pC3 s−T BsL T t ! q1 sds e−qC3 s−T AsL 3.42 T ! with Lt : eC3 t−T Lt. Now we apply Lemma 2.3 to get ! Lt ≤ N −q −q t −1/q ! Asds , 3.43 t ≥ T, 0 ∞ −1/p ! ! . If, in addition, where At : e−qC3 t−T At and N : LT −p − p 0 Bsds ∞ −q ! ! q 0 Asds < LT , then Lt is uniformly bounded by a positive constant C4 . Thus Lt ≤ C4 e−C3 t , 3.44 t ≥ T, and by continuity 3.44 holds for all t ≥ 0. If limt → ∞ ηt 0, then for any C > 0 there exists a tC ≥ t∗ such that ηt ≤ C for t ≥ tC. Therefore, L t ≤ −C5 ηtLt BtLp1 t AtLq1 t, 3.45 t ≥ t tC2 t for some C5 > 0. The previous argument carries out with eC3 t−T replaced by eC5 T ηdds . 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 h ≤ −Ch 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 < 2q 1λ1 − αat and b t < p 2λ1 − βbt may be relaxed to a t < 2q 1λ1 at and b t < p 2λ1 bt, respectively. In this case α αt and β βt would depend on t. 16 Journal of Applied Mathematics 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. 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