Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 901631, 14 pages
doi:10.1155/2011/901631
Research Article
Oscillation Properties for Second-Order
Partial Differential Equations with Damping and
Functional Arguments
Run Xu, Yuhua Lu, and Fanwei Meng
Department of Mathematics, Qufu Normal University, Shandong, Qufu 273165, China
Correspondence should be addressed to Run Xu, xurun 2005@163.com
Received 19 September 2011; Accepted 21 October 2011
Academic Editor: Norio Yoshida
Copyright q 2011 Run Xu et al. 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.
Using an integral averaging method and generalized Riccati technique, by introducing a parameter
β ≥ 1, we derive new oscillation criteria for second-order partial differential equations with damping. The results are of high degree of generality and sharper than most known ones.
1. Introduction
Consider the second-order partial delay differential equation
s
∂ux, t
∂
∂
atΔux, t ak tΔu x, t − ρk t − qx, tfux, t
rt ux, t pt
∂t
∂t
∂t
k1
−
m
qj x, tfj u x, t − σj ,
x, t ∈ Ω × R ≡ G,
j1
1.1
where Δ is the Laplacian in RN , R 0, ∞ and Ω is a bounded domain in RN with a piecewise smooth boundary ∂Ω.
Throughout this paper, we assume that
H1 rt ∈ C1 R , 0, ∞, pt ∈ CR , R;
H2 qx, t, qj x, t ∈ CG, R , qt minx∈G qx, t, qj t minx∈G qj x, t, j ∈ Im {1,
2, . . . , m};
2
Abstract and Applied Analysis
H3 at, ak t, ρk t ∈ CR , R , limt → ∞ t − ρk t ∞, and σj are nonnegative constants, j ∈ Im , k ∈ Is {1, 2, . . . , s};
H4 fu ∈ C1 R, R, fj u ∈ CR, R are convex in R with ufj u > 0, ufu > 0, and
0.
f u ≥ μ > 0, u /
We say that a continuous function Ht, s belongs to the function class ω, denoted by
H ∈ ω, if H ∈ CD, R , where D {t, s : −∞ < s ≤ t < ∞}, satisfy
Ht, t 0,
Ht, s > 0,
−∞ < s < t < ∞.
1.2
Furthermore, the continuous partial derivative ∂H/∂S exists on D, and there is h ∈
Lloc D, R, such that
∂H
−ht, s Ht, s.
∂s
1.3
Various results on the oscillation for the partial functional differential equation have
been obtained recently. We refer the reader to 1–3
for parabolic equations and to 4–11
for
hyperbolic equations.
Recently, Li and Cui 12
studied the equation of the form
l
s
∂
∂
atΔux, t ak tΔu x, t − ρk t −qtux, t
pt
ux, t λi tux, t − τi ∂t
∂t
i1
k1
−
m
qj x, tu x, t − σj ,
x, t ∈ Ω × R ≡ G
j1
1.4
with Robin boundary condition
∂ux, t
gx, tux, t 0,
∂γ
x, t ∈ ∂Ω × R ,
1.5
where γ is the unit exterior vector to ∂Ω and gx, t is a nonnegative continuous function on
∂Ω × R and obtained the following result.
Theorem A see 12, Theorem 2.2
. Suppose that H ∈ ω, let
C1 0 < infs≥t0 {lim inft → ∞ Ht, s/Ht, t0 } ≤ ∞, suppose that there exists some j0 ∈ Im
and there exist two functions φ ∈ C1 t0 , ∞, A ∈ Ct0 , ∞ satisfying,
t
C2 lim supt → ∞ 1/Ht, t0 t0 ps − σj0 φsh2 t, sds < ∞,
∞
C3 t0 A2 s/ps − σj0 φsds ∞, and for every t1 ≥ t0 ,
t
C4 lim supt → ∞ 1/Ht, t1 t1 Ht, sψs − 1/4φsps − σj0 h2 t, s
ds ≥ At1 ,
s
where φs exp{−2 φξdξ}, A s max{As, 0}, and ψs φs{αj0 qj0 s 1− li1 λi s−
σj0 ps − σj0 φ2 s − ps − σj0 φs
}. Then every solution ux, t of the problem 1.4, 1.5 is
oscillatory in G.
Abstract and Applied Analysis
3
In 2008, Rogovchenko and Tuncay 13
established new oscillation criteria for secondorder nonlinear differential equations with damping term
rtx t ptx t qtfxt 0,
1.6
without an assumption that has been required in related results reported in the literature over
the last two decades. Motivated by the ideas in 12, 13
, by introducing a Parameter β ≥ 1,
we will further improve Theorems A and derive new interval criteria for oscillation of 1.1.
We suggest two different approaches which allow one to remove condition C2 in Theorem
A. A modified integral averaging technique enables one to simplify essentially the proofs of
oscillation criteria.
2. Main Results
Theorem 2.1. Suppose that there exists a function y ∈ C1 t0 , ∞ such that for some β ≥ 1 and for
some H ∈ ω,
lim sup
t→∞
1
Ht, t0 t Ht, sψs −
t0
β
vsrsh2 t, s ds ∞,
4μ
t ≥ 0,
2.1
where
t
ps
ds ,
vt exp −2
μys −
2rs
ψt vt qt μrty2 t − ptyt − rtyt ,
2.2
2.3
then every solution ux, t of the problem 1.1, 1.5 is oscillatory in G.
Proof. Suppose to the contrary that there is a nonoscillatory solution ux, t of the problem
1.1, 1.5 which has no zero on Ω × t0 , ∞ for some t0 > 0. Without loss of generality, we
assume that ux, t > 0, ux, t−ρk t > 0 and ux, t−σj > 0 in Ω×t1 , ∞, t1 ≥ t0 , k ∈ Is , j ∈ Im .
Integrating 1.1 with respect to x over the domain Ω, we have
d
d
d
rt
ux, tdx pt
ux, tdx
dt
dt Ω
dt Ω
s
at
Δux, tdx ak t
Δu x, t − ρk t dx
Ω
−
Ω
qx, tfux, tdx −
2.4
Ω
k1
m j1
Ω
qj x, tfj u x, t − σj dx,
t ≥ t1 .
4
Abstract and Applied Analysis
From Green’s formula and the boundary condition 1.5, we have
Ω
Ω
Δux, tdx Δu x, t − ρk t dx ∂Ω
∂Ω
∂ux, t
ds −
gx, tux, tds ≤ 0,
∂γ
∂Ω
∂u x, t − ρk t
ds
∂γ
2.5
g x, t − ρk t u x, t − ρk t ds ≤ 0,
−
t ≥ t1 , k ∈ I s ,
∂Ω
where ds denotes the surface element on ∂Ω. Moreover, from H2 , H4 and Jensen’s inequality, we have
qx, tfux, tdx ≥ qt
Ω
Ω
qj x, tfj u x, t − σj dx ≥ qj t
Ω
fux, tdx ≥ |Ω|qtf
Ω
Ω
Ω
ux, tdx ,
fj u x, t − σj dx
≥ |Ω|qj tfj
where |Ω| Set
1
|Ω|
1
|Ω|
Ω
2.6
u x, t − σj dx,
dx.
1
Ut |Ω|
Ω
ux, tdx,
t ≥ t1 .
2.7
In view of 2.5–2.7, 2.4 yields that
m
rtU t ptU t qtfUt qj tfj U t − σj ≤ 0,
t ≥ t1 .
2.8
j1
Note that H4 , 2.8 yields that
rtU t ptU t qtfUt ≤ 0,
t ≥ t1 .
2.9
Put
U t
yt ,
wt vtrt
fUt
t ≥ t1 ,
2.10
Abstract and Applied Analysis
5
where vt is given by 2.2, then
2
pt
rtU t rtf UtU t
wt vt
−
rtyt
w t −2μyt rt
fUt
f 2 Ut
2
pt
rtU t μrtU t
≤ −2μyt wt vt
−
rtyt
rt
fUt
f 2 Ut
ptU t
pt
U t 2 wt − vt
qt μrt
≤ −2μyt − rtyt
rt
fUt
fUt
pt
wt
−2μyt wt − vt pt
− yt qt
rt
vtrt
2
wt
μrt
− yt − rtyt
vtrt
pt
−2μyt wt
rt
− vt
pt
wt− ptytqt
vtrt
wtyt
w2 t
μrt 2
μrty2 t − rtyt
−2μ
2
vt
v tr t
−2μytwt pt
pt
wt −
wt − vt −ptyt qt μrty2 t − rtyt
rt
rt
2μytwt − μ
w2 t
vtrt
w2 t
−vt −ptyt qt μrty2 t − rtyt − μ
vtrt
−ψt − μ
w2 t
,
vtrt
2.11
that is,
ψt ≤ −w t − μ
w2 t
,
vtrt
2.12
6
Abstract and Applied Analysis
where ψt is defined by 2.3. Multiplying 2.12 by Ht, s and integrating from T to t, we
have, for some β ≥ 1 and for all t ≥ T ≥ t1 ,
t
Ht, sψsds ≤ −
t
T
Ht, sw sds −
T
Ht, T wT −
⎡
⎣
T
β
4μ
t
μ
w2 sds
vsrs
μw2 s
ht, s Ht, sws Ht, s
vsrs
T
Ht, T wT −
Ht, s
T
t t
t
μHt, s
ws βvsrs
vsrsh t, sds −
2
ds
⎤2
βvsrs
ht, s⎦ ds
4μ
2.13
t T
T
β−1 μ
Ht, sw2 sds.
βvsrs
Writing the latter inequality in the form
t T
β
vsrsh2 t, s ds
4μ
⎤2
⎡
t μHt, s
βvsrs
ws ht, s⎦ ds
≤ Ht, T wT − ⎣
βvsrs
4μ
T
Ht, sψs −
−
2.14
t T
β−1 μ
Ht, sw2 sds.
βvsrs
Using the properties of Ht, s, we have
t β
vsrsh2 t, s ds ≤ Ht, t1 |wt1 | ≤ Ht, t0 |wt1 |,
Ht, sψs −
4μ
t1
t ≥ t1 ,
2.15
and for all t ≥ t1 ≥ t0 ,
t t0
t
1
β
2
ψs ds |wt1 | .
vsrsh t, s ds ≤ Ht, t0 Ht, sψs −
4μ
t0
2.16
By 2.16,
lim sup
t→∞
1
Ht, t0 t t1
β
ψsds |wt1 | < ∞,
vsrsh2 t, s ds ≤
Ht, sψs −
4μ
t0
t0
2.17
which contradicts 2.1. This proves Theorem 2.1.
Abstract and Applied Analysis
7
Consider a Kamenev-type function Ht, s defined by Ht, s t − sn−1 , t, s ∈ D,
where n > 2 is an integer. Obviously, H belongs to the class ω, and ht, s n − 1t −
sn−3/2 , t, s ∈ D. Then, we can get the following results.
Corollary 2.2. Suppose that there exists a function yt ∈ C1 t0 , ∞; R such that for some integer
n > 2 and some β ≥ 1,
lim sup
t→∞
1
t
tn−1
βn − 12
vsrs ds ∞,
t − s ψs −
4μ
t − s
n−3
t0
2
2.18
where vt and ψt are as defined in Theorem 2.1. Then every solution ux, t of the problem 1.1,
1.5 is oscillatory in G.
Theorem 2.3. Suppose that
Ht, s
0 < inf lim inf
≤ ∞.
s≥t0 t → ∞
Ht, t0 2.19
Assume that there exist functions f ∈ C1 t0 , ∞; R and φ ∈ Ct0 , ∞; R such that, for all t ≥ T ≥
t0 and for some β > 1,
1
lim sup
t→∞
Ht, T t β
2
vsrsh t, s ds ≥ φT ,
Ht, sψs −
4μ
T
2.20
where vt, ψt are as defined in Theorem 2.1 and suppose further that
t
lim sup
t→∞
t0
φ2 s
∞,
vsrs
2.21
where φ t maxφt, 0. Then every solution ux, t of the problem 1.1, 1.5 is oscillatory in G.
Proof. Suppose to the contrary that there is a nonoscillatory solution ux, t of the problem
1.1, 1.5 which has no zero on Ω × t1 , ∞ for some t1 > t0 , without loss of generality, we
assume that ux, t > 0, ux, t − ρk t > 0 and ux, t − σj > 0 in Ω × t1 , ∞, t ≥ t1 ≥ t0 , k ∈
Is , j ∈ I m .
8
Abstract and Applied Analysis
As in the proof of Theorem 2.1, 2.14 holds for all t ≥ T ≥ t1 , we have
t β
vsrsh2 t, s ds
Ht, sψs −
4μ
T
⎤2
⎡
t μHt, s
βvsrs
1
⎣
ws ht, s⎦ ds
≤ wT −
Ht, T T
βvsrs
4μ
1
Ht, T t β−1 μ
1
Ht, sw2 sds
−
Ht, T T βvsrs
t β−1 μ
1
Ht, sw2 sds.
≤ wT −
Ht, T T βvsrs
2.22
Therefore, for t > T ≥ T0 ,
t→∞
t β
vsrsh2 t, s ds
4μ
T
t β−1 μ
1
Ht, sw2 sds.
≤ wT − lim inf
t→∞
Ht, T T βvsrs
lim sup
1
Ht, T Ht, sψs −
2.23
It follows from 2.20 that
1
wT ≥ φT lim inf
t→∞
Ht, T t T
β−1 μ
Ht, sw2 sds
βvsrs
2.24
for all T ≥ t1 and for any β > 1. Then, for all T ≥ t1 ,
wT ≥ φT ,
lim inf
t→∞
1
Ht, t1 t
t1
β
Ht, s 2
w sds ≤ wt1 − φt1 .
vsrs
μ β−1
2.25
2.26
Now, we claim that
∞
t1
w2 s
ds < ∞.
vsrs
2.27
w2 s
ds ∞.
vsrs
2.28
Suppose the contrary, that is,
∞
t1
Abstract and Applied Analysis
9
By 2.19, there is a positive constant M1 , satisfying
Ht, s
inf lim inf
s≥t0 t → ∞
Ht, t0 > M1 > 0.
2.29
Let M be any arbitrary positive number, then from 2.28 we get that there exists a T1 > t1
such that, for all t ≥ T1 ,
t
t1
M
w2 s
ds ≥
.
vsrs
M1
2.30
Using integration by parts, for all t ≥ T1 , we get
1
Ht, t1 s
w2 τ
dτ ds
t1
t1 vτrτ
t s
1
w2 τ
∂Ht, s
≥
dτ ds
−
Ht, t1 T1
∂s
t1 vτrτ
t
t w2 s
1
ds Ht, s
vsrs
Ht,
t1 t1
M
1
≥
M1 Ht, t1 ∂Ht, s
−
∂s
t T1
∂Ht, s
ds
−
∂s
2.31
M Ht, T1 .
M1 Ht, t1 By 2.29, there exists a T2 > T1 such that, for all t ≥ T2 ,
Ht, T1 ≥ M1 .
Ht, t1 2.32
It follows from 2.31 that for all t ≥ T2 ,
1
Ht, t1 t
Ht, s
t1
w2 s
ds ≥ M.
vsrs
2.33
Since M is an arbitrary positive constant,
lim inf
t→∞
1
Ht, t1 t
Ht, s
t1
w2 s
ds ∞,
vsrs
2.34
which contradicts 2.26. Consequently, 2.27 holds. And from 2.25, we obtain
∞
t1
φ2 s
ds ≤
vsrs
∞
t1
w2 s
ds < ∞,
vsrs
which contradicts 2.21. This completes the proof of Theorem 2.3.
2.35
10
Abstract and Applied Analysis
Choosing H as in Corollary 2.2, by Theorem 2.3, we can obtain the following corollary.
Corollary 2.4. Let vt and ψt be as in Theorem 2.1, assume further that there exist functions
f ∈ C1 t0 , ∞; R and φ ∈ Ct0 , ∞; R such that, for all T ≥ t0 , for some integer n > 2, and for
some β > 1,
lim sup
t→∞
1
tn−1
t
βn − 12
vsrs ds ≥ φT t − s ψs −
4μ
t − s
n−3
T
2
2.36
and 2.21 hold. Then every solution ux, t of the problem 1.1, 1.5 is oscillatory in G.
Theorem 2.5. Suppose that there exists a function f ∈ C1 t0 , ∞; R such that for some β ≥ 1 and
for some H ∈ ω,
1
lim sup
t→∞
Ht, t0 t μβ
p2 svs
2
Ht, s ψs −
−
vsrsh t, s ds ∞,
2μrs
2
t0
2.37
where
vt exp −2μ
t
ysds ,
ψt vt qt μrty2 t − ptyt − rtyt .
2.38
Then every solution ux, t of the problem 1.1, 1.5 is oscillatory in G.
Proof. As in Theorem 2.1, without loss of generality, we assume that a nonoscillatory solution
ux, t of the problem 1.1, 1.5 satisfies ux, t > 0, ux, t − ρk t > 0 and ux, t − σj > 0 in
Ω × t1 , ∞, t1 ≥ t0 , k ∈ Is , j ∈ Im . Define a generalized Riccati transformation
U t
yt ,
wt vtrt
fUt
t ≥ t1 ,
2.39
where vt is given by 2.38. Then
2 wt
wt
− yt −μrt
−yt
w t ≤ −2μytwtvt −qt rtyt −pt
vtrt
vtrt
−ψt −
pt
1
wt − μ
w2 t,
rt
rtvt
t ≥ t1 .
2.40
Using an elementary inequality
a
b2
,
−ax2 bx ≤ − x2 2
2a
2.41
Abstract and Applied Analysis
11
for all a > 0 and for all b, x ∈ R, we conclude from 2.40 that
ψt −
p2 tvt
μ
≤ −w t −
w2 t,
2μrt
2vtrt
2.42
t ≥ t1 .
Multiplying 2.42 by Ht, s and integrating from T < t, we obtain, for some β ≥ 1 and for all
t ≥ T ≥ t1 ,
t
p2 svs
Ht, s ψs −
2μrs
T
ds ≤ Ht, T wT −
t
≤ Ht, T wT −μ
t
μw2 s
ds
ht, s Ht, swsds−
T
T 2vsrs
μβ
2
t
vsrsh2 t, sds
T
t T
μ
−
2
β − 1 Ht, s 2
w sds
2βvsrs
t T
2
Ht, s
ws βvsrsht, s ds.
βvsrs
2.43
Therefore, for all t ≥ T ≥ t1 , we have
t T
p2 svs μβ
2
Ht, sψs − Ht, s
−
vsrsh t, s ds
2μrs
2
≤ Ht, T wT − μ
t T
μ
−
2
t T
β − 1 Ht, s 2
w sds
2βvsrs
Ht, s
ws βvsrsht, s
βvsrs
2.44
2
ds.
Following the same lines as in the proof of Theorem 2.1, we have
1
lim sup
t→∞
Ht, t0 ≤
t1
p2 svs μβ
2
Ht, sψs − Ht, s
−
vsrsh t, s ds
2μrs
2
t t0
ψsds |wt1 | < ∞
t0
which contradicts the assumption 2.19.
This completes the proof.
2.45
12
Abstract and Applied Analysis
Theorem 2.6. Let 2.19 holds. Assume that there exist functions f ∈ C1 t0 , ∞; R and φ ∈
Ct0 , ∞, R such that, for all t ≥ t0 , any T ≥ t0 , and for some β > 1,
1
lim sup
t→∞
Ht, T t T
p2 svs
Ht, s ψs −
2μrs
μβ
2
−
vsrsh t, s ds ≥ φT 2
2.46
and 2.21 holds, where ψt, vt are defined as in Theorem 2.6 and φ t maxφt, 0, then every
solution ux, t of the problem 1.1, 1.5 is oscillatory in G.
Theorem 2.7. Let all assumptions of Theorem 2.6 be satisfied except that condition 2.46 be replaced
by
1
lim inf
t→∞
Ht, T t μβ
p2 svs
2
Ht, s ψs −
−
vsrsh t, s ds ≥ φT .
2μrs
2
T
2.47
Then every solution ux, t of the problem 1.1, 1.5 is oscillatory in G.
Remark 2.8. By introducing the parameter β in Theorem 2.3, we derive new oscillation
criteria of the problem 1.1, 1.5 which are simpler than that in Theorem A; furthermore,
modifications of the proofs through the refinement of the standard integral averaging method
allowed us to shorten significantly the proofs of Theorem 2.3. We can also derive a number
of oscillation criteria with the appropriate choice of the function H and ρ, here, we omit the
details.
3. Examples
Now, we consider these following examples.
Example 3.1. Consider the partial differential equation
∂ux, t
∂ 1 ∂
1
ux, t ux, t − π 2 cos t
∂t t ∂t
2
∂t
1
3
Δux, t 2 Δu x, t − π
2
t
2
1
− 3 t cos2 t sin t fux, t − 2 f1 ux, t−π,
t
t
3.1
x, t ∈ 0, π×0, ∞,
with the boundary condition
ux 0, t ux π, t 0,
t > 0,
3.2
where fu u3 u, f1 u ueu u.
Here, N 1, l 1, s 1, m 1, μ 1, rt 1/t, pt 2 cos t, qx, t qt 2/t3 2
t cos t sin t, q1 x, t 1/t2 , fu fj u u, at 1, a1 t 1/t2 , ρ1 t 3/2π, σ1 π, τ1 π.
Abstract and Applied Analysis
13
Let
yt 1
t cos t,
t
3.3
then
vt t2 ,
ψt t−1 .
3.4
Let n 3, for any β ≥ 1,
lim sup
t→∞
1
t2
t
t − s2 s−1 − βs2
1
1
1 t
ds lim sup 2
t − s2 s−1 − βs ds ∞.
t→∞
s
t 1
3.5
Therefore, Corollary 2.2 holds, then every solution ux, t of the problem 3.1, 3.2 oscillates
in 0, π × 0, ∞.
Example 3.2. Consider the partial differential equation
∂
∂t
∂ux, t
∂ux, t
1
3
1
1
sin
t
2
2 sin t
∂t
t
∂t
2t3
2t3
3
3Δux, t 2 − cos tΔu x, t − π − t−3 1 − t3 2t2 − 6t sin t 12t fux, t
2
π − 2t sin tf1 ux, t − π − 2f2 u x, t −
, x, t ∈ 0, π × 0, ∞
2
3.6
1
with the boundary condition 3.2, where fu u5 u, f1 u u sin2 u, f2 u u3 cos2 u.
Here N 1, s 1, m 2, μ 1, rt 1 1/2t3 2 sin t, pt 3/t1 1/2t3 2 sin t, qt t−3 1 − t3 2t2 − 6t sin t 12t
, at 3, a1 t 2 − cos t, q1 x, t 2 sin t,
q2 x, t 2, ρ1 t 3/2π, σ1 π, σ2 π/2.
Let yt 0, then vt t3 and ψt vtqt 1 − t3 2t2 − 6t sin t 12t.
Choose β 2, n 3, a straightforward computation yields
1
lim sup 2
t→∞ t
t t − s2
1 − s3 2s2 − 6s sin s 12s − 2s3 1 2 sin s ds
T
3.7
16 − T 3 cos T T 2 2 cos T − 6 3 sin T − 4T sin T − 3 cos T φT .
Let φ t maxφt, 0. It is not difficult to see that
t
lim sup
t→∞
1
φ2 s
ds ≥ lim sup
s3 1/22 sin s
t→∞
t
1
φ2 s
ds ∞.
3s3 1/2
3.8
By Corollary 2.4, we obtain that every solution of problem 3.6, 3.2 oscillates in 0, π ×
0, ∞.
14
Abstract and Applied Analysis
Note that in this example,
lim sup
t→∞
1
t2
t 1
4 s3 2 sin s ds ∞,
2
1
3.9
so the condition C2 would not have been satisfied with the same choices of vt.
Acknowledgments
This research was supported by National Science Foundation of China 11171178, the fund of
subject for doctor of ministry of education 20103705110003, and the Natural Science Foundations of Shandong Province of China ZR2009AM011 and ZR2009AL015.
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