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, tatarn@kfupm.edu.sa
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
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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
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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
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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
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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
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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.
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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.
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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.
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