Proyecciones Journal of Mathematics 10.22199/issn.0717-6279-4357-4415
Vol. 40, No 6, pp. 1615-1639, December 2021.
Universidad Católica del Norte
Antofagasta - Chile
On asymptotic behavior of solution to a
nonlinear wave equation with Space-time speed
of propagation and damping terms
Paul A. Ogbiyele
University of Ibadan, Nigeria
and
Peter O. Arawomo
University of Ibadan, Nigeria
Received : August 2020. Accepted : August 2021
Abstract
In this paper, we consider the asymptotic behavior of solution to
the nonline¡ar damped¢wave equation
utt − div a(t, x)∇u + b(t, x)ut = −|u|p−1u t ∈ [0,∞), x ∈ Rn
u(0, x) = u0(x), ut(0, x) = u1(x) x ∈ Rn
with space-time speed of propagation and damping potential. We ob-
tained L2 decay estimates via the weighted energy method and under
certain suitable assumptions on the functions a(t, x) and b(t, x). The
technique follows that of Lin et al.[8] with modification to the region
of consideration in Rn. These decay result extends the results in the
literature.
Subjclass Primary: 35L05, 35L70; Secondary: 37L15
Keywords: Space-time speed of propagation, Space-time dependent
damping, Asymptotic behavior, Weighted energy method.
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1616 Paul A. Ogbiyele and Peter O. Arawomo
1. Introduction
In this paper, we are concerned with the asymptotic behavior of solution
to the following nonlinear wave equation
( ³ ´
u − div a(t, x)∇u + b(t, x)u = −|u|p−1tt t u, t ∈ [0,∞), x ∈ Rn
u(0, x) = u0(x), ut(0, x) = u1(x) x ∈ Rn,
(1.1)
with space-time dependent coefficients of the form
−α
(1.2) b(t, x) = b0(1 + |x|2) −β2 (1 + t)
and
ρ1(1+|x|2
δ
) 2 (1+t)γ |ξ|2 ≤ a(t, x)ξ·ξ ≤ ρ (1+|x|2 δ0 ) 2 (1+t)γ |ξ|2, ξ ∈ Rn
(1.3)
where a(t, x) = η(t)−1ρ(x) and η(t) = (1 + t)−γ . In addition, b0 > 0,
ρ0 > 0, α+ δ ∈ [0, 2) and β + γ ∈ [0, 1), where u = u(t, x). More precisely,
α + β + δ + γ ∈ [0, 1). Equations of the form (1.1) arise in the study of
nonlinear wave equations describing the motion of body traveling in an
in-homogeneous medium. They appear in various aspects of Mathematical
Physics, Geophysics and Ocean acoustics.
In the case of scalar coefficients and bounded smooth domains Ω, there
is an extensive literature on energy dacay results. For the semi-linear wave
equation
(1.4) utt −∆u+ ut = |u|p,
Todorova and Yordanov [18] showed that C 2n = 1 + n is the critical
exponent(Fujita exponent) for p <∞ (n < 3) and p < 1 + 2n(n ≥ 3).
Nishihara in his paper [11] showed that the decay rate of solution to
the damped linear wave equation follows that of self similar solution of
its corresponding heat equation for n = 3 and showed this by obtaining
Lp − Lq estimates on their difference. For similar results on 1-dimension
and 2-dimensions, see Marcati and Nishihara [9] and Hosono and Ogawa [5]
respectively, and in any dimension, see Narazaki [10]. Hence, it is expected
that the behavior of the solution to equation (1.4) is similar to that of the
corresponding heat equation
(1.5) ut −∆u = |u|p,
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On asymptotic behavior of solution to a nonlinear wave equation ..1.617
−1 1
whose similarity solution ua(t, x) has the form tp−1F (xt
−
2 ) with
2
a = lim 2|x|→∞ |x| p−1 f(x) ≥ 0 provided that p < 1 + n .
In the case of time dependent potential type of damping, with equations
of the form
(1.6) utt −∆u+ b(t)u + |u|p−1t u = 0,
there are also several results on the decay rate of the solution. Nishihara
and Zhai [13], used a weighted energy method similar to those in [18] and
obtained decay estimates of the form
n
kuk2 ≤ −( )(1+β)Ct 4(p−1)(1.7)
k k ≤ −(
n )(1+β)
u Ct 2(p−1)1
under the assumption that b(t) ≈ (1 + t)−β. For Cauchy problem of the
form
(1.8) u − a2tt (t)∆u+ b(t)ut + c0|u|p−1u = 0,
it is well known that the interplay between the coefficient a2(t) and the term
b(t)ut induces different effect on the asymptotic behavior of the energy E(t)
given by
1 a2(t)
(1.9) E(t) = ku 2tk + k∇uk2
1
2 + kukpp.2 2 p
For more details see [2, 3, 4, 20] and the references therein. In [1] Bui
considered the asymptotic behavior of the nonlinear problem (1.8) with
a(t) = (1+ t) and b(t) = µ(1+ )(1+ t)−1, > 0, c0 = 0 and obtained the
following estimate
³ ´
ku +( +1)max{µ∗− 1 ,−1}t(t, ·), (1+t) ∇u(t, ·)kL2 ≤ (1+t) 2 ku1kH1+ku2kL2
(1.10)
with µ∗ = 12(1− µ− +1).
In the case of damped wave equation with space dependent potential
type of damping;
(1.11) utt −∆u+ b(x)u p−1t + |u| u = 0,
where b1(1 + |x|)−α ≤ b(x) ≤ b2(1 + |x|)−α and b1, b2 > 0, Todorova and
Yordanov [19] investigated the decay rate of the energy when 0 ≤ α < 1.
They obtained several decay rate types for solutions of (1.11) depending on
p and α. These decay rates take the form
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1618 Paul A. Ogbiyele and Peter O. Arawomo
³ ´ ³ −1 − p+1 ´
(1.12) ku k + k∇uk , kuk = O t +δp−1 , t 2(p− +δ1)t 2 2 p+1
if 1 < p < 1 + 2αn−α and
³ ´ ³ α 1 n α p+1 n ´
k k k∇ k k k −(1+ ) − + +δ −(1+ ) + +δu + u , u = O t 2 p 1 2(p+1) 2 2(p−1) 4t 2 2 p+1 , t
(1.13)
if 1 + 2α 2(4−α)n−α < p < 1 + (n−α)(4−α) , for t > 1, where δ is a constant.
Nishihara[12] also considered the asymptotic behavior of solution to the
semi-linear wave equation (1.11) with b(x) satisfying
| |2 −α(1.14) b1(1 + x ) 2 ≤
α
b(x) ≤ b 2 −2(1 + |x| ) 2
and obtained decay rates of the following type
⎧⎪⎪⎪⎪ − n−2α⎨ C(1 + t) 2(2−α) if 1 + 2 n+2n−α ≤ p < n−22 1 n
k C(1 + t)
− ( )−
2−α p−1 4 if 1 + 2α < p ≤ 1 + 2
u(t, ·)k n−α n−α2 ≤ ⎪⎪⎪⎪ − 2 ( 1 )−n 1⎩ C(1 + t) 2−α p−1 4 [log(t+ 2)] 2 if p = 1 + 2αn−α− 1 + α
C(1 + t) p−1 2(2−α) if 1 < p < 1 + 2αn−α
(1.15)
where α ∈ [0, 1).
Ikehata and Inoue [6] studied nonlinear wave equations with b(x) = b0(1 +
|x|)−1 and showed that solutions to (1.11) depend on the coefficient b0 and
their decay estimate takes the form
(1.16) kuk = O(t−1+µ) kutk22 + k∇uk2 = O(t−1+µ2 )
where
1 < µ+ b0 < 1 + b0 if 0 < b0 ≤ 1
0 ≤ µ < 1 if b0 ≥ 1.
Moreover, for damped wave equations with space-time dependent po-
tential type of damping
utt −∆u+ b(t, x)ut + |u|p−1u = 0, t > 0, x ∈ Rn(1.17)
u(0, x) = u n0(x), ut(0, x) = u1(x), x ∈ R ,
Lin et al. [8] considered decay rates of solution to (1.17) and showed
using the weighted energy method that the L2 norm of the solution decays
as
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On asymptotic behavior of solution to a nonlinear wave equation ..1.619
⎧⎪⎪⎪⎨ −( 1 − α )(1+β)C(1 + t) p−1 2(2−α) if α(p+1)p−1 > n
ku(t.·)k2 ≤ ⎪⎪⎩⎪
−( 1 α
C(1 + t) p−
−
1 2(2− )(1+β)α) log(t+ 2), if α(p+1)p−1 = n
−(1+β) 1 + 1+β− − (N−α
2 )
C(1 + t) p 1 2(2 α) p−1 if α(p+1)p−1 < n
(1.18)
For nonlinear wave equations with variable coefficients which exhibit a
dissipative term with a space dependent potential
(1.19) utt −∇ · (b(x)∇u) +∇ · (b(x)ut) = 0, x ∈ Rn, t > 0
under the assumption that
(1.20) b0(1 + |x|)β|ξ|2 ≤ b(x)ξ · ξ ≤ b (1 + |x|)β|ξ|21 , ξ ∈ Rn,
where b0 > 0, b1 > 0 and β ∈ [0, 2). R. Ikehata et al. [7] obtained long time
asymptotics for solutions to (1.19)-(1.20) as a combination of solutions of
wave and diffusion equations under certain assumptions on b in an exterior
domain, see also [15].
Said-Houari [17] considered a viscoelastic wave equation with space-
time dependent damping potential and an absorbing term
R
u ttt −∆u+ 0 g(t− s)∆u(s)ds+ b(t, x)u + |u|p−1t u = 0, t > 0, x ∈ Rn
u(0, x) = u0(x), u
n
t(0, x) = u1(x) x ∈ R
(1.21)
and by using a weighted energy method, they showed that the L2 decay
rates are the same as those in [8].
More recently, Roberts[16] under the assumption that
b0(1+|x|)β ≤ b(x) ≤ b1(1+|x|)β and a0(1+|x|)−α ≤ a(x) ≤ a1(1+|x|)−α
with
(1.22) α < 1, 0 ≤ β < 2, 2α+ β < 2,
obtained energy decay estimates of solution to the dissipative non-linear
wave equation
u − div(b(x)∇u) + a(x)u + |u|p−1u = 0, x ∈ Rntt t , t > 0(1.23)
u(0, x) = u0(x) ∈ H1(Rn), ut(0, x) = u1(x) ∈ L2(Rn),
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1620 Paul A. Ogbiyele and Peter O. Arawomo
using a modification of the weighted multiplier technique introduced by
Todorova and Yordanov[14].
In this paper, by using the weighted L2-energy method similar to that of
[8], we obtain decay estimates of the energy of the solution to (1.1), where
a(t, x) and b(t, x) have the form in (1.2)-(1.3) above. In [8], the space Rn
was divided into two zones
Z(t;L, t n 2 20) := {x ∈ R |(t0 + t) ≥ L+ |x| }
and Zc(t;L, t ) = Rn0 \Z(t;L, t0). To obtain boundedness on certain esti-
mates on Z, a further division of Z was required. Here, we split the domain
into two zones
Ω(t, L, t0) = {x ∈ Rn : (t + t)A0 ≥ L+ |x|2} and
Ωc(t, L, t0) = R
n\Ω(t, L, t0)
which depend on the weighted function for A = 2(1+β+γ)2−(α+δ) and positive
constants L, t0. With this choice, we overcome the challenge of splitting the
first zone in order to obtain boundedness for every estimate on Ω(t;L, t0)
in the proof.
2. Preliminaries
In this section, we state some basic assumptions used in this paper. First,
we introduce the following notations. Lp(Rn), 1 ≤ p ≤ ∞, the Lebesgue
space with norm k · k and H1p ρ(Rn)Zthe Sobolev space defined by
(2.1) H1 n
2n
ρ(R ) := {u ∈ Ln−2+δ : (1 + |x|2
δ
) |∇u|22 dx <∞}.
Rn
Lemma 2.1. (Caffarelli-Kohn-Nirenberg)
There exist a constant C > 0 such that the inequality
(2.2) k|x|σukLr ≤ Ck|x|δ∇ukθ 1−θLqk|x| ukLp
holds for all u ∈ C∞(Rn0 ) if³and only if the following relations hold:1 σ 1 δ − 1´ ³1 ´
(2.3) + = θ + + (1− θ) +
r n q n p n
with p, q ≥ 1. r > 0, 0 ≤ θ ≤ 1. δ − d ≤ 1 if θ > 0 and 1 + δ−1p n =
1
r +
σ
n
Remark 1. When σ = δ = = 0, the Lemma is referred to as the
Gagliardo-Nirenberg inequality.
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On asymptotic behavior of solution to a nonlinear wave equation ..1.621
We define the weighted function ψ(t, x) as follows:
2−(α+δ)
(L+ |x|2) 2
(2.4) ψ(t, x) = λ
(t0 + t)1+β+γ
for a small positive constant λ = b0(1+β+γ)2ρ (2−(α+δ))2 and t0 ≥ L ≥ 1. Moreover,0
we have
2−(α+δ)
2 2
ψt(t, x) = −λ(1 + β + γ) (L+|x| )(t0+t)2+β+γ
−α−δ
∇ − (L+|x|
2) 2ψ(t, x) = λ(2 (α+ δ)) x
(t0+t)1+β+γ
2 −α−δ 2
|∇ψ(t, x)|2 = λ2(2− (α+ δ))2 (L+|x| ) |x|
(t +t)2+2β+2γ0
and consequently, we have
a(t, x)|∇ψ|2 1
(2.5) − ≤ b(t, x).( ψt(t, x)) 2
In the sequel, we will denote the function ψ(t, x) by ψ for simplicity.
To begin, we state the following lemmas which will be needed in the proof
of the main result. First, we define the functions E(t) and H(t) associated
to problem (1.1) by
h i
(2.6) E(t) := e2ψη(t) 1 |u |2 + a(t,x)2 t 2 |∇u|2 +
1 p+1
p+1 |u|
and
h
H 2ψ b(t, x)
i
(2.7) (t) := e η(t) uut + |u|2
2
respectively. Then for the function E(t) in (2.6), we have the following
result.
Lemma 2.2. Let u be a solution of (1.1), then the function E(t) defined
in (2.6), satisfies
h i
d E(t) ≤ ∇ · (e2ψρh(x)∇uu ) + e2ψη(it) − b(t,x) + ψh |u |2i+ e2ψ ηt(t)dt t 4 t t 2 |u 2t|
+e2ψη(t) −γ + 2ψt |u|p+1(p+1)(1+t) p+1 + e2ψ
ρ(x)ψt
3 |∇u|2.
(2.8)
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1622 Paul A. Ogbiyele and Peter O. Arawomo
Proof. Multiplying (1.1) by e2ψut and using (2.5), we obtain
∙ h i¸
d e2ψ 1 |u |2 + a(t,x)dt 2 t 2 |∇u|2 +h 1p+1 |u|p+1 i
2ψ
= ∇ · (e2ψha(t, x)∇uu 2ψ 2 e at(t,x) 2t) + ei ψt − b(t, x) |ut| + 2 |∇u|
(2.9) e2ψa(t,x)
2h e2ψa(t,x)|∇ψi|2+ ψ |∇u|2 −∇ψu − |u |2 + 2e2ψψt p+1ψ t t t |u|t ψt p+1
2ψ
≤ ∇ · (e2ψha(t, x)∇uu 2ψ 1 2 e at(t,x) 2t) + ei ψt − 2b(t, x) |ut| + 2 |∇u|
+e
2ψa(t,x) 2 2ψψ 2e ψt p+1ψ t|∇u|−∇ψut + p+1 |u| ,t
where we have used
(2.10) e2ψut · b(t, x)u = e2ψb(t, x)|u |2t t .
By employing Schwartz inequality, we observe that
h i
e2ψa(t,x) 2
ψ ψt|∇u|−h∇ψutt i
2ψ
(2.11) = e a(t,x)ψ h|ψt|2|∇u|2 − 2ψ 2 2tut∇u ·∇ψi+ |∇ψ| |ut|t
≤ e
2ψa(t,x) 1 |ψ |2ψ t |∇u|2 −
1 |∇ψ|2|u 2t| .
t 3 2
Hence, using (2.5) in (2.11) and substituting the resulting estimate in
(2.9), we obtain
∙ h i¸
d e2ψ 1 |u |2 + a(t,x) |∇u|2 +h 1 |u|p+1dt 2 t 2 p+1 i
(2.12) ≤ ∇ · (eh2ψa(t, x)∇uu ) +i e2ψ ψ − b(t,x) |u |2 + 2e2ψψt |u|p+1t t 4 t p+1
+e2ψ at(t,x) + a(t,x)ψt2 3 |∇u|2
and multiplying (2.12) by η(t), we get
∙ h i¸
d
dt e
2ψη(t) 12 |u |2 +
a(t,x) 2
t 2 |∇u| h+ 1 |u|p+1p+1 i
≤ ∇ · (e2ψρh(x)∇uu ) + e2ψη(it) − b(t,x)t 4 + ψh 2 2ψ ηt(t) 2t |ut|i+ e 2 |ut|
+e2ψη(t) −γ + 2ψt |u|p+1 + e2ψ ρ(x)ψt(p+1)(1+t) p+1 3 |∇u|2.
(2.13)
2
Now, for the function H(t), we have the following lemma.
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On asymptotic behavior of solution to a nonlinear wave equation ..1.623
Lemma 2.3. Let u be a solution of (1.1), then the function H(t) defined
in (2.7), satisfies
d
dtH(t) ≤ ∇ · (e2ψρ(x)u∇u) + e2ψη∙(t)|u |2 + 2e2ψt η¸(t)ψtuut − e2ψη(t)|u|p+1
e2ψ− ρ(x) |∇u|2 + e2ψ4 η(t)
bt(t,x) b(t,x)ψt 2
2 + 3 |u|
+e2ψ ηt(t)b(t,x)2 |u|2 + e2ψηt(t)uut
(2.14)
Proof. Multiplying (1.1) by e2ψu and using the estimate (2.5), we get
∙ h i¸
d e2ψ uu + b(t,x)dt t 2 |u|2
= ∇ · (e2ψa(t, x)u∇u) + e2ψ|u |2 + 2e2ψt ψtuut + e2ψ bt(t,x)2 |u|2
2
−e2ψa(ht, x)|∇u|2 − a (t,x)|∇ψ|2 2 2ψψ b(t,xi) |∇u| e − e2ψ|u|p+1(2.15) t
+ b(t,x) |ψ u+ a(t,x)∇ψ
2
ψ t b(t,x) |∇u| e2ψt
≤ ∇ · (e2ψa(t, x)u∇u) + eh2ψ|u |2 + 2e2ψψ uu i+ e2ψ bt(t,x) 2t t t 2 |u|
e2ψ− a(t,x)
2
2 |∇u|2 +
b(t,x) |ψ u− a(t,x)∇ψ |∇u| e2ψ − e2ψ|u|p+1ψ tt b(t,x)
where we have used ∙ ¸
2ψ
e2ψb(t, x)uut =
d e b(t,x)
dt 2 |u|2 − e2ψψ 2tb(t, x)|u|(2.16)
−e2ψ bt(t,x)2 |u|2.
Using Schwartz inequality for the second to the last term on the right
hand side of (2.15), we hhave the following esh i
timate
b(t,x) a(t,x)∇ψ 2
(2.17) ψ
|ψtu+
t b(t,x)
|∇u| i
b(t,x) 1 |a(t,x)|2|∇ψ|2≤ 2 2 2ψt 3 |ψt| |u| − 2|b(t,x)|2 |∇u| .
In a similar way, using (2.5) in (2.17), and substituting the resulting
estimate in (2.15), we get
∙ h i¸
d e2ψ uu + b(t,x) 2dt t 2 |u|
(2.18)≤ ∇ · (e2ψa(t, x)u∇u) + e2ψ|u |2 + 2e2ψψ uu + e2ψ bt(t,x) 2t t t 2 |u|
2ψ
−e a(t,x) |∇u|2 + e2ψ b(t,x)ψt |u|2 − e2ψ|u|p+14 3
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1624 Paul A. Ogbiyele and Peter O. Arawomo
and multiplying (2.18) by η(t), we obtain
∙ h i¸
d e2ψη(t) uu + b(t,x) |u|2dt t 2
≤ ∇ · (e2ψρ(x)u∇u) + e2∙ψη(t)|u |2 + 2e2¸ψη(t)ψ uu − e2ψη(t)|u|p+1t t t
e2ψ− ρ(x) |∇u|2 + e2ψη(t) bt(t,x) + b(t,x)ψt |u|24 2 3
+e2ψ ηt(t)b(t,x) |u|22 + e2ψηt(t)uut.
(2.19)
2
3. Main result
In this section, we consider the long time behavior of the solution to (1.1).
The result here is obtained via a weighted energy method and the technique
follows that of Lin et al.[8]. For local existence result, the compactness
condition on the support of the initial data is replaced by the following
condition:
Z ∙
αA h i ¸β+
I := η(0) t 2 2 2 2 2ψ(0,x)0 Z 0
|u1| + a(0, x)|∇u0| + b(0, x)|u0| e dx
Ω(0;L,t0) ∙ h i ¸
1 αA
+ η(0) (L+ |x|2) (β+ )A 2 |u 2 2 21| + a(0, x)|∇u0| + b(0, x)|u0|
Ωc(0;L,t0)
e2ψ(0,x)dx < +∞.
(3.1)
With respect to the size of (1 + |x|2) and (1 + t) and considering the
weighted function ψ, we partition the space Rn into the following zones:
Ω(t, L, t0) = {x ∈ Rn : (t0 + t)A ≥ L+ |x|2} and
Ωc(t, L, t0) = R
n\Ω(t, L, t0)
which is a modification of the zones as inspired by Lin et. al. [8], where
A = 2(1+β+γ)2−(α+δ) . Since α+ β + δ + γ ∈ [0, 1), it follows that A < 2.
Theorem 3.1. Let u be the solution of (1.1) and let a(t, x), b(t, x) satisfy
(1.2) and (1.3) for 2 < p+1 < 2nn−2+δ when n ≥ 2. Suppose that (u0, u1) ∈
H1 n 2ρ(R ) ∩ L (Rn) and (??) holds. Then there exist a unique solution u
of (1.1) with u ∈ L∞([0,∞);H1ρ(Rn)) and u ∈ L∞t ([0,∞);L2(Rn)) which
satisfies the following estimate
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On asymptotic behavior of solution to a nonlinear wave equation ..1.625
⎪⎪⎪⎧⎨ −2(1+β) α(1+β+γ)+C(1 + t) p−1 2−(δ+α) , if α(p+1)(p−1) > n
⎪⎪ −2(1+β) α(1+β+γ)(3.2)kuk2 ≤ +⎪ C(1 + t) p−1 2−(δ+α) log(2 + t), if
α(p+1)
L2 ⎩ (p−1) = n−2(1+β)+ 1+β+γ (n− 2α )C(1 + t) p−1 2−(δ+α) p−1 , if α(p+1)(p−1) < n.
Remark 2. The existence result can be proved using the same technique
as in [8] where in this case the Caffarelli-Kohn-Nirenberg inequality is used
instead of the Gagliardo-Nirenberg inequality, with the additional consid-
δ
eration of the inequality |x|δ ≤ (1 + |x|2) 2 . Hence, we omit the proof here.
Proof. [Proof of Theorem 3.1] We split the proof into three parts, the
first part considers the case x ∈ Ω(t, L, t0), the second part covers the case
x ∈ Ωc(t, L, t0) and the third part combines the two results . We state the
result in each of the zones in the form of a lemma.
Case 1: (x ∈ Ω(t, L, t0)). In this region, we define a functionEψ(Ω(t, L, t0))
by
αA
(3.3) Eψ(Ω(t, L, t
β+
0)) := (t0 + t) 2 E(t) + νH(t)
where ν is a small positive constant to be determined later, and the func-
tions HE(t;Ω(t;L, t0)), H1(t) and H2(t) by
R
(3.4)HE(t;Ω(t;L, t0)) := Ω(t;L,t )Eψ(Ω(t, L, t0))dx0
R ¯̄̄ h iN−1
H (t) := 2π1 0 Eψ( Ω(t,qL, t A 20)) √ (t + t) − L dθ|x| 0= (t0+t)A−L
× ddt (t0 + t)A − L
(3.5) Z h i
αA
(3.6) H (t) := e2ψ (t + t)β+2 0 2 ρ(x)∇uut + νρ(x)u∇u ·−→n dS
∂Ω(t;L,t0)
where −→n is the unit outward normal vector of ∂Ω(t;L, t0). Then we state
the next lemma.
IBADAN UNIVERSITY LIBRARY
1626 Paul A. Ogbiyele and Peter O. Arawomo
Lemma 3.2. Let u be a solution of (1.1) and the functions E(t) and H(t)
be defined as in (2.6) and (2.7) above, then for x ∈ Ω(t, L, t0), the function
Eψ(Ω(t, L, t0)) satisfies
d
dtEψ(Ω(t,hL, t0)) i
≤ ∇ · (e2ψ ∙ αA(t0 + t)β+ 2 ρ(x)∇uut +¸µνρ(x)u∇u ) ¶
(3.7)−k ∙e2ψ ¸ αA0 η(t) 1 + (t0 + t)β+ 2 (−ψt) |ut|2 + a(t, x)|∇u|2 + |u|p+1
−k 10 (t +t) + (−ψ 2ψt) e η(t)b(t, x)|u|2 − k0e2ψη(t)|u|p+10
where k0 is a positive constant to be determined later. Furthermore, we
have
³ i ³ ´
d
dt (t⎧ m⎪⎪0 + t) HE(t;Ω(t;L, t0)) − (t0 + t)
m H1(t) +H2(t)
⎪⎨ − − (1+β)(p+1)C(1 + t)m γ p−1 , if α(p+1)(p−1) > n(3.8) ⎪⎪ (1+β)(p+1)≤ ⎪ C(1 + t)m−γ− p−1 log(2 + t), if α(p+1)⎩ (p−1) = n− − (1+β)(p+1) 1+β+γ −α(p+1)m γ + (n )C(1 + t) p−1 2−(δ+α) p−1 , if α(p+1)(p−1) < n.
αA
Proof. Multiplying (2.8) by (t + t)β+0 2 , we obtain
∙ ¸
d αA
dt (t0 + t)
β+
2 E(t)
≤∙∇ · αA αA(e2ψ(t0 + t)β+ ρ(x)∇uu ) + ηt(t)2 t 2 (t0 + t)β+ 2 |ut¸|2
αA
(3.9)+∙ (β+ ) − b(t,x) β+αA β+αA2 (t 2ψαA 4 0 + t) ¸2 + (t0 + t) 2 ψt e η(t)|u |22(t +t)1−(β+ ) t0 2
αA
+∙ (β+ )2 ψt β+αA 2ψ 2αA + 3 (t0 + t) 2 e ρ(x)|∇u|2(t0+t)1−(β+ 2 ) ¸
(β+αA )−γ 2ψ β+αA+ 2 t− αA + p+1(t0 + t) 2 e
2ψη(t)|u|p+1.
(p+1)(t +t)1 (β+ 2 )0
Observe that β + αA2 ≤ β + α < 1 since A < 2 and α+ β + δ + γ < 1.
Now, multiplying (2.14) by ν (where ν < b0) and adding the resulting
estimate to (3.9), we get
IBADAN UNIVERSITY LIBRARY
On asymptotic behavior of solution to a nonlinear wave equation ..1.627
∙ ¸
d αA(t0 + t)hβ+ 2dt E(t) + νH(t) i
≤∙∇ · αA(e2ψ (t0 + t)β+ 2 ρ(x)∇uut + νρ(x)u∇u ) ¸
(β+αA )−γ(1− ν )
+∙ 2 b0 + ν − b0 + ( 1b0−3ν) β+αAαA 4 b (t0 + t) 2 ψt e2ψη(t)|ut|22(t0+t)1−(β+ 2 ) 1 0(3.10) ¸
∙ (β+
αA ) − ν¸ ψ β+αA+ 2 + t (t0 + t) 2αA 4 3 e2ψρ(x)|∇u|22(t0+t)1−(β+ 2 )
+∙ν −β + (1− 1)2(t +t) 3 ψ e2ψt η(t)b(t, x)|u|20 ¸
(β+αA )−γ − 2ψ αA+ 2 tαA ν + p+1(t + t)
β+
2 e2ψ0 η(t)|u|p+1,
(p+1)(t 1−(β+ )0+t) 2
where we have used Schwartz inequality to obtain the following estimates
for the third and last term on the right hand side of (2.14) respectively:
| α2ψ u u| ≤ 1b(t,x)(−ψt) |u|2 + 3(−ψt)t t 3 b (1 + t)β(1 + |x|2) 2 |u 2t|(3.11) 1 0
≤ − 1b(t,x)ψt |u|2 − 3ψ β+αAt 2 23 1b (t0 + t) |ut|0
and
| | ≤ −b(t,x)η αη (t)u u t(t) |u|2 − ηt(t) β 2t t 22 2b (1 + t) (1 + |x| ) |u 2t|(3.12) 0
≤ −b(t,x)η αAt(t)2 |u|2 −
ηt(t)
2b (t + t)
β+
2 |u |20 t .
0
By a suitable choice of ν sufficiently small as mentioned earlier, we
can now choose a positive constant k0 such that the estimates below are
satisfied
(β+αA )−γ(1− ν )
2 b0 + ν − b0αA 4 ≤ −k1−(β+ ) 0
2t 20
αA αA
(3.13) (β+ )2
− αA −
ν ≤ − (β+ )−γ4 k0, 21 (β+ ) 1−(β+αA − ν ≤ −2k) 0
2t 20 (p+1)t
2
0
ν 1− 1 ≥ k , ( 1b0−3ν) 1 23 0 b ≥ k0, 3 ≥ k0, (p+1) ≥ k0, ν
β ≥ k0,
1 0 2
this gives the desired estimate (3.7).
We now integrate the estimate (3.7) over Ω(t;L, t0) to obtain
d
(3.14) HE(t;Ω(t;L, t0))−H1(t)−H2(t) ≤ −H3(t;Ω(t;L, t0)),
dt
IBADAN UNIVERSITY LIBRARY
1628 Paul A. Ogbiyele and Peter O. Arawomo
where
H3 (t;Ω(tZ;L, t0)) ∙
2ψ αA αA:= k0 e η(t) (1 + (−ψt)(t0 + t)β+ 2 )|ut|2 + (1 + (−ψ )(t + t)β+t 0 2 )
Ω(t;L,t0)
a(³t, x)|∇u|2 ´ ¸
αA
+ −ψt + 1t +t b(t, x)|u|2 + (1 + (−ψ )(t + t)
β+
2 )|u|p+1 + |u|p+1t 0 dx.
0
(3.15)
Define the function HE by
Z
HE(t;Ω(t;L, t0)) := η(t)
(3.16)∙ h Ω(t;L,t0) i ¸
β+αA(t0 + t) 2 |ut|2 + a(t, x)|∇u|2 + |u|p+1 + b(t, x)|u|2 e2ψdx.
It can be proved easily that for positive constants k1, k2, the following
inequality is satisfied:
(3.17) k1HE ≤ HE(t;Ω(t;L, t0)) ≤ k2HE .
Now, multiplying (3.14) by (t0 + t)
m for m a constant which will be
determined later, we obtain
³ i ³ ´
d m m
dt (t0 + t) H∙E(t;Ω(t;L, t0)) − (t0 + t) H1(t) +H2((3.18) ¸t)
≤ (t + t)m m0 t +tHE(t;Ω(t;L, t0))−H3(t;Ω(t;L, t0)) .0
The term on the right hand side is estimated as
m
t +t HE(t;Ω(t;L, t0))−H3(t;Ω(t;L, t0))0
≤ mZk2t +tHE(t;Ω(t;∙L, t0))−H3(t;Ω(t;L0 ¸h, t0)) i
≤ e2ψη(t) mk2 − k |u |2αA 0 t + a(t, x)|∇u|2 + |u|p+1 dxZ ∙(ht +t)1i−(β+ )0 2Ω(t;L,t0) ¸
+ e2ψη(t) mk2 2 p+1t0+t b(t, x)u − k0|u| dx,
Ω(t;L,t0)
(3.19)
IBADAN UNIVERSITY LIBRARY
On asymptotic behavior of solution to a nonlinear wave equation ..1.629
where we have used ψt ≤ 0.
From (3.13), it can be easily seen that we can choose t0 large enough, such
that mk2 k0− αA < 2 . Therefore, the first term on the right hand side of1 (β+ 2 )t0
(3.19) yields
Z ∙ ¸h i
e2ψη(t) mk2 − k |u |2 + a(t, x)|∇u|2 + |u|p+1 dx
(tZ+t)1−(β+αA ) 0 t0 2Ω(t;L,t0)
≤ −k0 e2ψη(t)(|u |2 + a(t, x)|∇u|2 + |u|p+12 t )dx ≤ 0.
Ω(t;L,t0)
(3.20)
To estimate the second term on the right hand of (3.19), we apply
Young’s inequality to obtain
Z ∙h i ¸
e2ψη(t) mk2 b(t, x)u2 − k |u|p+1t +t 0 dx0
Ω(t;L,t0) Z ∙h i ¸
≤ 2ψ mk −αe η(t) 21+β b0(1 + |x|2) 2 |u|2 − k0|u|p+1 dxZ ∙ (1+t)Ω(t;L,t0) ¸−(1+β)(p+1) −α(p+1)
≤ e2ψη(t) C(1 + t) p−1 (1 + |x|2) 2(p−1) − kp|u|p+1 dx
Ω(t;L,t0)
− (1+β)(p+1) R −α(p+1)≤ Cη(t)(1 + t) p−1 e2ψ(1 + |x|2) 2(p−1) dx
Ω(t;L,t0)
−α(p+1)
− (1+β)(p+1) R A ³ ´≤ − (t0+t) 2Cη(t)(1 + t) 1 + r2 2(p−1)p 1 0 rn−1dr
(3.21)
where C = C(m, b0, k2, p) and kRp = kp(k³0, p). D´efine J byA −α(p+1)− (1+β)(p+1)− −γ (t +t) 2J := C(1 + t) 0 1 + r2 2(p−1)p 1 rn−10 dr.
Thus, if α(p+1)(p−1) > n, it follows that
− (1+β)(p+1)(3.22) J ≤ C(1 + t) p− −γ1 ,
if α(p+1)(p−1) = n, we have
(3.23) J ≤ C(1 + t)−
(1+β)(p+1)
− −γp 1 log(2 + t)
and if α(p+1)(p−1) < n, we obtain
IBADAN UNIVERSITY LIBRARY
1630 Paul A. Ogbiyele and Peter O. Arawomo
(1+β)(p+1)
(3.24) ≤ − − −γ+
1+β+γ α(p+1)
− (n− )J C(1 + t) p 1 2 (δ+α) p−1 .
Combining (3.19) - (3.24), we have
m
t +⎧⎪⎪tHE(t;Ω(t;L, t0))−H3(t;Ω(t;L, t0))0 ⎨⎪⎪ (1+β)(p+1)C(1 + t)− − −γp 1 , if α(p+1)(p−1) > n(3.25) ⎪⎪ − (1+β)(p+1)≤ ⎪⎪ C(1 + t) p−
−γ
1 log(2 + t), if α(p+1)⎩ (p−1) = n− (1+β)(p+1)− −γ+ 1+β+γ α(p+1)− (n− )C(1 + t) p 1 2 (δ+α) p−1 , if α(p+1)(p−1) < n.
Hence, we have that
³ i ³ ´
d m m
dt (t⎪⎪⎧0 + t) HE(t;Ω(t;L, t0)) − (t0 + t) H1(t) +H2(t)⎪⎪ − − (1+β)(p+1)C(1 + t)m γ p−1 , if α(p+1)(p−1) > n
(3.26) ⎨⎪⎪ − − (1+β)(p+1)≤ ⎪⎪ C(1 + t)
m γ
p−1
⎩ log(2 + t), if
α(p+1)
(p−1) = n
− − (1+β)(p+1)m γ + 1+β+γ −α(p+1)
C(1 + t) p−1 2−
(n
(δ+α) p− )1 , if α(p+1)(p−1) < n.
n o 2
Case 2: For the region Ωc(t;L, t0) = x|(t + t)A0 ≤ L + |x|2 , we define
another function Eψ(Ω
c(t, L, t0)) by
1 αA
(3.27) E cψ(Ω (t, L, t0)) := (L+ |x|2) (β+ )A 2 E(t) + νH(t),
where ν is a small positive constant to be determined later. In addition,
define
R
H (t;ΩcE (t;L, t0)) :=
c
Ωc(t;L,t )Eψ(Ω (t, L, t0))dx0
(3.28)
¯̄̄ h iN−1
H∗
R
1 (t) :=
2π
0 Eψ( Ω
c(tq, L, t0)) √ (t0 + t)A − L 2 dθ|x|= (t +t)A0 −L
× ddt (t A0 + t) − L
(3.29)
IBADAN UNIVERSITY LIBRARY
On asymptotic behavior of solution to a nonlinear wave equation ..1.631
Z h i
H∗
1 αA
2 (t) := e
2ψ (L+|x|2) (β+ ) −→A 2 ρ(x)∇uut+νρ(x)u∇u · n dS
∂Ωc(t;L,t0)
(3.30)
where −→n is the unit outward normal vector of ∂Ωc(t;L, t0).
We can now state the next lemma.
Lemma 3.3. Let u be a solution of (1.1) and the functions E(t) and H(t)
be defined as in (2.6) and (2.7) above, then for x ∈ Ωc(t;L, t0), the function
E (Ωcψ (t, L, t0)) satisfies
d
dtE (Ω
c
ψ (th, L, t0)) i
≤ ∇ · (e2ψ ∙ 1 αA(L+ |x|2) (β+ )A 2 ρ(x)∇uut +¸µνρ(x)u∇u ) ¶
− 2ψ 1 αAk0∙e η(t) 1 + (L+¸ |x|2) (β+ )A 2 (−ψt) |ut|2 + a(t, x)|∇u|2 + |u|p+1
−k 1 + (− 1ψ ) e2ψη(t)b(t, x)|u|2 − k [1 + (L+ |x|2)− [1−(β+αA )]0 A 2(t +t) t 0 ]0
e2ψη(t)|u|p+1
(3.31)
where k0 is a positive constant to be determined later. Moreover, we have
that
h i ³ ´
d (t + t)mH (t;Ωc(t;L, t )) − (t + t)mdt 0 E 0 0 H1(t) +H2(t) ≤ 0.
(3.32)
1 αA
Proof. Multiplying (2.8) by (L+ |x|2) (β+ )A 2 , we obtain
∙ ¸
d 1(L+ |x|2) (β+αA )A 2dt E(t)
≤ ∇ ·∙ 1 αA 1 αA(e2ψ(L+ |x|2) (β+ )A 2 ρ(x)∇uut) + e2ψ ηt(t)2 (L+ |¸x|2) (β+ )A 2 |ut|2
+η∙(t) − b(t,x)(L+ |x|2 1) (β+αA ) 1 (β+αA )A¸ 24 + (L+ |x|2)A 2 ψ 2ψt e |ut|2
1 αA
+ (L+ |x|2 1) (β+αA )ψA 2 t 2ψh 3 e ρ(x)|∇u|
2 − (β+ )A 2 2ψ
1− 1 e x · ρ(x)∇uu(β+αA ) t
(L+|x|2) A 2
1
− | |2 (β+
αA i
2ψ γ(L+ x )A 2
)
2ψ 1 αA+e η(t) t(p+1)(1+t) + p+1(L+ |x|2)
(β+ )
A 2 |u|p+1.
(3.33)
IBADAN UNIVERSITY LIBRARY
1632 Paul A. Ogbiyele and Peter O. Arawomo
Adding (3.33) to ν× (2.19), we obtain
d
dtEψ(Ω
c(th, L, t0)) i
≤ ∇ · 1(e2ψ (L+ |x|2) (β+αA )A 2 ρ(x)∇uut + νρ(x)u∇u )
− 1 αA 2ψ 2 1 (β+αA )−1A 2A(β∙+ 2 )e (L+ |x| ) x · ρ(x)∇uut + νe2ψ η¸t(t)b(t,x)2 |u|2
+η∙(t) ν − b(t,x) | |2 1(L+ x ) (β+¸αA )A 2 + (L+ |x|2 1 αA4 ) (β+ )A 2 ψt e2ψ|u 2t|
+ −ν 2 1 (β+αA ) ψ 2ψ 2 2ψ η (t) 2 1t t (β+αA )A 2 A 2 24∙+ (L+ |x| ) 3 e ρ(x)|∇u| + e 2 ¸(L+ |x| ) |ut|1 αA
2 (β+ )A 2 1 αA
+η(∙t) −ν − γ(L¸+|x| ) + 2ψt (L+ |x|2) (β+ )A 2 2ψ p+1(p+1)(1+t) p+1 e |u|
+ν −β + ψt 2ψ2(t +t) 3 e η(t)b(t, x)|u|2 + 2νe2ψη(t)ψtuut + νe2ψηt(t)uut.0
(3.34)
For the second term on the right hand of (3.34), by using Schwartz
inequality, we obtain
| 1 αA | |2 1(β + )(L+ x ) (β+αA )−1A 2A 2 x · ρ(x)∇uut|
≤ 1 1 αA 1A(β +
αA
2 )(L+ |x|2)
(β+ )−
A 2 2 |ut|ρ(x)|∇u|
1 (β+αA(3.35) ≤ )ρ(x)
1 (β+αA )
A 2
− 1 αA ρ(x)|∇u|
2 + A 2 2
1 (β+1+ ) 1 αA
|ut|
2(L+|x|2) A 2 2(L+| |2 [1−(β+x )A 2 )]
1
≤ (β+
αA )ρ 10 2 (β+
αA )
A 2 A 2 2
− 1 (α+δ)A
ρ(x)|∇u| + 1 [1−(β+αA |u)] t|
2(L+|x|2)1 (β+1+ 2 )A 2(L+|x|2)A 2
and observe here that 1 (β + 1 + (α+δ)A) = 2(β+1)+γ(α+δ)A 2 2(1+β+γ) < 1. Also, by
using the Schwartz inequality, we obtain the following estimates for the
second to the last term and the last term on the right hand side of (3.34)
respectively:
|2ψ uu | ≤ 2 (−ψ )b(t, x)|u|2 + 3 (−ψ )(1 + t)β 2 αt t t t (1 + |x| ) 2 |u |2t
(3.36) 3 2b0
≤ − 23 (ψ )b(t, x)|u|2 −
3
t b (ψt)(L+ | |2
1 (β+αAx ) )A 2 |u 2t|
2 0
and
|η (t)u u| ≤ b(t,x)(−ηt(t)) |u|2 + (−ηt(t)) αt t 2 2b (1 + t)β(1 + |x|2) 2 |u 2t|(3.37) 0
≤ −b(t,x)η (t) | |2 − η (t) | |2 1 (β+αAt t )A 2 22 u 2b (L+ x ) |ut| .0
Therefore, substituting the estimates (3.35) - (3.37) in (3.34), we get
IBADAN UNIVERSITY LIBRARY
On asymptotic behavior of solution to a nonlinear wave equation ..1.633
d
dtEψ(Ω
c(th, L, t0)) i
≤ ∇ ·∙(e2ψ 1 αA(L+ |x|2) (β+ )A 2 ρ(x)∇uut + νρ(x)u∇u )1 (β+αA ¸∙ )−γ(1−
ν )
+η(t) ν + A 2 b0 − b0 + (1− 3ν )(L+¸|x|2 1 αA) (β+ )A 2 2ψ1 ψ e |u 2[1−(β+αA )] 4 b t t|2LA 2 2 0
1
−ν∙ (β+
αA )ρ0 | |2 1 (β+αA+ 4 + A 2 (α+δ)A + (L+ x )
)ψ
A 2 t e2ψ3 ρ(x)|∇u|21− 1 (β+1+ 2 )2L A ¸
+η(∙t) − − ¸ γ 2ψ 2 1 (β+αAν t )A 2 2ψ p+11 αA + p+1(L+ |x| ) e |u|[1−(β+ )](p+1)(L+|x|2)A 2
+ν −β + (1− 2)ψ e2ψ2(t +t) 3 t η(t)b(t, x)|u|2.0
(3.38)
Now, just as in the Case 1, we choose a suitable value for ν which is
sufficiently small and a positive constant k0 such that the estimates we have
below are satisfied.
1 (β+αA )−γ(1− ν ) 1 (β+αA
ν + A 2 b0 − b0 ≤ −k ν )ρA 2 01 αA 4 0, − 4 + 1 (α+δ)A ≤ −k0,
2L [1−(β+ 2 )]A 2L1− (β+1+ )A 2
ν (1− 2)3 ≥ k0,
2
p+1 ≥ k0,
1
3 ≥ k0, (1−
3ν
b ) ≥ k0, ν ≥ 2k0,2 0
βv ≥ k , γ2 0 p+1 ≥ k0,
(3.39)
which gives the desired estimate. Therefore by integrating the estimate
(3.31) over Ωc(t, L, t0), we obtain
d
(3.40) HE(t;Ω
c(t;L, t0))−H∗1 (t)−H∗2 (t) ≤ −H (t;Ωc3 (t;L, t0))dt
where
H3(t;Ω
c(Zt;L, t0)) ∙h i
1 αA
:= k η(t)e2ψ 1 + (−ψ )(L+ |x|2) (β+ )
h 0
A 2
(3.41) Ωc(t;L,t0) i t
|u³ 2t| + a(t, x)´|∇u|2 + |u|p+1 ¸
− 1 αA+ ψt + 1t +t b(t, x)|u|2 + [1 + (L+ |x|2)
− [1−(β+ )]
A 2 ]|u|p+1 dx
0
Define the function H cE by
IBADAN UNIVERSITY LIBRARY
1634 Paul A. Ogbiyele and Peter O. Arawomo
H cE Z ∙ h i ¸
| |2 1 (β+αA= η(t) (L+ x ) )A 2 |u |2t + a(t, x)|∇u|2 + |u|p+1 + b(t, x)|u|2 e2ψdx.
Ωc(t;L,t0)
(3.42)
It can be proved in a similar way as in Case 1 that for positive constants
k∗ ∗1, k2, the following inequality holds.
(3.43) k∗1H cE ≤ HE(t;Ωc(t;L, t0)) ≤ k∗H c2 E .
Multiplying (3.40) by (t0 + t)
m for the same constant m as in Case 1,
we have
h i ³ ´
d m c m ∗ ∗
dt (t0 + t) H
(3.44) ∙E(t;Ω (t;L, t0)) − (t0 + t) H1 (t) +H2 (t¸)
≤ (t + t)m m H c c0 t +t E(t;Ω (t;L, t0))−H3(t;Ω (t;L, t0)) .0
The term on the right hand side is estimated as
m
t +t HE(t;Ω
c(t;L, t0))−H3(t;Ωc(t;L, t0))
0
≤ mkZ∗2H ct +t E −H∙ c3(t;Ω (t;L, t0))0 ¸
∗ 1| |2 (β+
αA ) h i
≤ 2ψ mk2(L+ x )A 2e 2 1 (β+αA )A 2(t0+t) − k0 1 + (−ψt)(L+ |x| )
Ωc(t;L,t0)
Z h i×η(t) |u∙|2µ+ a(t, x)|∇u|2 +¶ |u|p+1t dx ¸
2ψ mk
∗
+ e η(t) 2t +t − k (−ψ 2 p+10 t) b(t, x)u − k0|u| dx.0
Ωc(t;L,t0)
(3.45)
It can be seen from (3.39) that we can suitably choose k0 such that
mk∗2 ≤ λk0(1 + β + γ). Therefore the first term on the right hand side of
(3.45) yields
IBADAN UNIVERSITY LIBRARY
On asymptotic behavior of solution to a nonlinear wave equation ..1.635
Z ∙
∗ 2−(δ+α)
¸
2ψ | |2 1 (β+αA mk
2
e (L + x ) )A 2 2(t +t) −
2
k0λ(1 + β + γ)
(L+|x| )
h 0 i (t0+t)2+β+γΩc(t;L,t0)
Z ×η(t) |u |2t + a(t, x∙)|∇u|2 + |u|p+1 dx1 ¸
(L+|x|2 (β+
αA )
≤ 2ψ )A 2e mk∗
h (t0+t) 2
− k0λ(1 + β + γ)
Ωc(t;L,t0) i
×η(t) |ut|2 + a(t, x)|∇u|2 + |u|p+1 dx ≤ 0.
(3.46)
Likewise, for the second term on the right hand side of (3.45), we have
Z ∙µ
∗ 2−(α+δ)
¶ ¸
2ψ mke η(t) 2 − k λ(1 + β + γ) (L+|x|
2) 2 2
t +t 0 2+β+γ b(t, x)u − k0|u|p+1 dx0 (t0+t)
Ωc(t;LZ,t0) ∙µ ¶ ¸
∗
≤ e2ψ mkη(t) 2 − k0λ(1+β+γ) 2t0+t (t0+t) b(t, x)u dx ≤ 0.
Ωc(t;L,t0)
(3.47)
Consequently, we have
h i ³ ´
(3.48d) (t + t)mdt 0 HE(t;Ω
c(t;L, t0)) − (t + t)m0 H∗1 (t) +H∗2 (t) ≤ 0.
2
Case 3. With t > L and H = H∗0 1 1 , H2 = H
∗
2 , then it follows from (3.26)
and (3.48) that
³ h i´
d m
dt (t⎪⎪⎧0 + t) H
c
E(t;Ω(t;L, t0)) +HE(t;Ω (t;L, t0))
⎨⎪ − − (1+β)(p+1)C(1 + t)m γ p−1 , if α(p+1)(p−1) > n(3.49) ⎪⎪ − − (1+β)(p+1)≤ ⎪ C(1 + t)m γ p−1 log(2 + t), if α(p+1)⎩⎪ (p−1) = n− − (1+β)(p+1)m γ + 1+β+γ α(p+1)− − (n− )C(1 + t) p 1 2 (δ+α) p−1 , if α(p+1)(p−1) < n.
Choosing
⎨⎧ (1+β)(p+1) α(p+1)
m = ⎩ p−1 − 1 + γ + if (p−1) > n(1+β)(p+1) 1+β+γ
p−1 − 2−(δ+α)(n−
α(p+1)
p−1 )− 1 + γ + if
α(p+1)
(p−1) < n,
(3.50)
IBADAN UNIVERSITY LIBRARY
1636 Paul A. Ogbiyele and Peter O. Arawomo
for 0 < < 1 and integrating (3.49) over [0, t], we obtain
h i
HE(t; Ω(⎪⎧⎪t;L, t0)) +HE(t;Ω
c(t;L, t0))
⎨⎪⎪ (1+β)(p+1)C(1 + t)− +1−γp−1 , if α(p+1)(p−1) > n⎪ (1+β)(p+1)≤ ⎪⎪⎩ C(1 + t)
− − +1−γp 1 log(2 + t), if α(p+1)(p−1) = n
− (1+β)(p+1)+ 1+β+γ −α(p+1)(n )+1−γ
C(1 + t) p−1 2−(δ+α) p−1 , if α(p+1)(p−1) < n.
(3.51)
In particulZar, we have Z
A := ⎧ e2ψb(t, x)|u|2dx+ e2ψb(t, x)|u|2dxΩ⎪⎪⎪(t;L,t
c
0) Ω (t;L,t0)
⎨ (1+β)(p+1)C(1 + t)− − +1p 1 , if α(p+1)(3.52) (p−1) > n⎪⎪ (1+β)(p+1)≤ ⎪⎪⎩ C(1 + t)
− +1
p−1 log(2 + t), if α(p+1)(p−1) = n
− (1+β)(p+1) 1+β+γ α(p+1)
C(1 + t) p−
+ (n− )+1
1 2−(δ+α) p−1 , if α(p+1)(p−1) < n.
2−(δ+α)
2 2
Now, set y = (L+|x| )1+β+γ . Since the following estimate(t0+t)
∙ 2−(δ+α) ¸ −α
2 2−(δ+α) −α
(1 + |x|2 −α −α 2 (1+β+γ)) 2 ≥ (L+ |x|2) = (L+|x| )2 (t + t) 2−(δ+α)
(t +t)1+β+γ 00
(3.53)
holds, then for y > 0, we have that
− α
(3.54) e2λyy 2−(δ+α) ≥ C.
Therefore, we obtain
Z
−β− α (1+β+γ)
(3.55) A ≥ C(1 + t) 2−(δ+α) u2dx
RN
which gives the desired estimate. 2
Remark 3. The decay result in Theorem 3.1 coincides with that of [8] for
the case δ = γ = 0 and with that of [13] for the case δ = γ = α = 0.
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On asymptotic behavior of solution to a nonlinear wave equation ..1.637
[1] TP.hB..  DN..  tBhuesi,i s",W Taevche models w
Riethfe trimene-cdeespendent speed and dissipation", 
[2] sTp. eBe.d N o.,  fB Mup. ir Roaupnzadhg 
nMic. aRl eUisnsivigersity Bergakademie Freiberg, 2013. 
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[3] Mwi. tDh ’Atibmbeic-cdoe paenndd eMn.t  Rs.p ,E evbeodel.r  3ta,9 n”9dA,   ndcolaa.m s1s,p  pionpfg . d”,i sJMan. aRly. sEisb aenrtd  aanpdp liWca.t iNo.n sNascimento, “A clas3si1fi5
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[4] with time-dependent  vmola. 2ss3 , annod.  1s1p, epepd.  8o4f7 -p8r8oc8pa,ta 2igo0an1t i8ofo.nr” ,w ave models Advances in Td.i fHfeorseonntial equations,[5] solutionso o afn 2d- dTi.m Oegnaswioan ,v a”olLl n.a 2rog0ne3l i,tn niemoa.er 1  bd, ea8hm2a-p1vei1od8r ,w aanvd
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[6] nwoit.1h,  pa pc. r1it3ic9a6l- 1p4o0te1n, t2i0al0 8ty. pe of damping”, near wave equations Nonlinear analysis, vol. 69, 
[7] Rdn.ao Im.k 8ep, hi npagpta . i,3 nG3 .H5 T2ilo-b3de3or6rt o8sv,p a2a,0 ca1en3sd”,  B. Yordanov, ”Wave equations witvho strong Journal of differential equations, l. 254, 
[8] Jw. aLvine,  eKq. uNaitsiohnihsa wrai,t ha nsdp aJc. eZ
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[10] applications top v−soeLlm.q 5i 6-el,is nntieoma. r2a t, eppsrp o.f ob5rl8e 5dm-a6”m,2 p6e, 2d0 w04av. e equations and their Journal of the Mathematical Society of Japan, 
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[11] eKq. uNatiisohnih ianr a3,-d”iLmpe−nsiLoqnaels tsipmaactee as nodf  thsoeilru taiponpsli ctaot iotnh”e,  damped wave Mathematische 
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[20] solution to nonli,n veoalr.  e2wtric 
7r,e pspo.n 3a1n7c-e3 37.6a0v, en eoq. 1u,a ptipo.n 2s5”1,a -n2d6 8n, o2n0e0x1is. tence of the global Journal of mathematical analysis and applications
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UNIVERSITY LIBRARY
On asymptotic behavior of solution to a nonlinear wave equation ..1.639
Paul A. Ogbiyele
Department of Mathematics,
University of Ibadan,
Ibadan, 200284,
Nigeria
e-mail: paulogbiyele@yahoo.com
Corresponding author
and
Peter O. Arawomo
Department of Mathematics,
University of Ibadan,
Ibadan, 200284,
Nigeria
e-mail: po.arawomo@ui.edu.ng
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