Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 27, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
OSCILLATION OF SOLUTIONS TO NEUTRAL NONLINEAR
IMPULSIVE HYPERBOLIC EQUATIONS WITH
SEVERAL DELAYS
JICHEN YANG, ANPING LIU, GUANGJIE LIU
Abstract. In this article, we study oscillatory properties of solutions to neutral nonlinear impulsive hyperbolic partial differential equations with several
delays. We establish sufficient conditions for oscillation of all solutions.
1. Introduction
The theory of partial differential equations can be applied to many fields, such as
to biology, population growth, engineering, generic repression, control theory and
climate model. In the last few years, the fundamental theory of partial differential equations with deviating argument has undergone intensive development. The
qualitative theory of this class of equations, however, is still in an initial stage of
development. Many srudies have been done under the assumption that the state
variables and system parameters change continuously. However, one may easily
visualize situations in nature where abrupt change such as shock and disasters may
occur. These phenomena are short-time perturbations whose duration is negligible in comparison with the duration of the whole evolution process. Consequently,
it is natural to assume, in modeling these problems, that these perturbations act
instantaneously, that is, in the form of impulses.
In 1991, the first paper [4] on this class of equations was published. However,
on oscillation theory of impulsive partial differential equations only a few of papers
have been published. Recently, Bainov, Minchev, Liu and Luo [1, 2, 6, 7, 8, 9, 10, 11]
investigated the oscillation of solutions of impulsive partial differential equations
with or without deviating argument. But there is a scarcity in the study of oscillation theory of nonlinear impulsive hyperbolic equations of neutral type with several
delays.
2000 Mathematics Subject Classification. 35B05, 35L70, 35R10, 35R12.
Key words and phrases. Oscillation; hyperbolic equation; impulsive; neutral type; delay.
c 2013 Texas State University - San Marcos.
Submitted October 1, 2012. Published January 27, 2013.
1
2
J. C. YANG, A. P. LIU, G. J. LIU
EJDE-2013/27
In this article, we discuss oscillatory properties of solutions for the nonlinear
impulsive hyperbolic equation of neutral type with several delays.
m
i
X
∂2 h
u(t,
x)
+
g
u(t
−
τ
,
x)
i
i
∂t2
i=1
= a(t)h(u)∆u − q(t, x)f (u(t, x)) +
l
X
ar (t)hr (u(t − σr , x))∆u(t − σr , x) (1.1)
r=1
−
n
X
qj (t, x)fj (u(t − ρj , x)),
t 6= tk , (t, x) ∈ R+ × Ω = G,
j=1
u(t+
k , x) = hk (tk , x, u(tk , x)),
ut (t+
k , x)
k = 1, 2, . . . ,
= pk (tk , x, ut (tk , x)),
k = 1, 2, . . . ,
(1.2)
(1.3)
with the boundary conditions
u = 0,
(t, x) ∈ R+ × ∂Ω,
∂u
+ ϕ(t, x)u = 0,
∂n
and the initial condition
u(t, x) = Φ(t, x),
(t, x) ∈ R+ × ∂Ω,
∂u(t, x)
= Ψ(t, x),
∂t
(1.4)
(1.5)
(t, x) ∈ [−δ, 0] × Ω.
Here Ω ⊂ RN is a bounded domain with boundary ∂Ω smooth enough and n is a
unit exterior normal vector of ∂Ω, δ = max{τi , σr , ρj }, Φ(t, x) ∈ C 2 ([−δ, 0] × Ω, R),
Ψ(t, x) ∈ C 1 ([−δ, 0] × Ω, R).
This article is organized as follows. in Section 2, we study the oscillatory properties of solutions for problems (1.1), (1.4). In Section 3, we discuss oscillatory
properties of solutions for problems (1.1), (1.5).
We will use the following conditions:
(H1) a(t), ai (t) ∈ P C(R+ , R+ ), τi , σr , ρj are positive constants, q(t, P
x), qj (t, x)
m
are functions in C(R+ × Ω̄, (0, ∞)), gi is a non-negative constant, i=1 gi <
1, i = 1, 2, . . . , m, j = 1, 2, . . . , n, r = 1, 2, . . . , l; where P C denote the class
of functions which are piecewise continuous in t with discontinuities of first
kind only at t = tk , k = 1, 2, . . . , and left continuous at t = tk , k = 1, 2, . . . .
(H2) h(u), hr (u) ∈ C 1 (R, R), f (u), fr (u) ∈ C(R, R); f (u)/u ≥ C a positive
constant, fj (u)/u ≥ Cj a positive constant, u 6= 0; h(0) = 0, hr (0) =
0, uh0 (u) ≥ 0, uh0r (u) ≥ 0, ϕ(t, x) ∈ C(R+ × ∂Ω, R), h(u)ϕ(t, x) ≥ 0,
hr (u)ϕ(t − σr , x) ≥ 0, r = 1, 2, . . . , l, 0 < t1 < t2 < · · · < tk < . . . ,
limk→∞ tk = ∞.
(H3) u(t, x) and their derivatives ut (t, x) are piecewise continuous in t with discontinuities of first kind only at t = tk , k = 1, 2, . . . , and left continuous at
−
t = tk , u(tk , x) = u(t−
k , x), ut (tk , x) = ut (tk , x), k = 1, 2, . . . .
(H4) hk (tk , x, u(tk , x)), pk (tk , x, ut (tk , x)) ∈ P C(R+ × Ω̄ × R, R), k = 1, 2, . . . ,
and there exist positive constants ak , ak , bk , bk and bk ≤ ak such that for
k = 1, 2, . . . ,
ak ≤
hk (tk , x, η)
≤ ak ,
η
bk ≤
pk (tk , x, φ)
≤ bk .
φ
EJDE-2013/27
OSCILLATION FOR HYPERBOLIC EQUATIONS
3
Let us construct the sequence {t̄k } = {tk } ∪ {tki } ∪ {tkr } ∪ {tkj }, where
tki = tk + τi , tkr = tk + σr , tkj = tk + ρj and t̄k < t̄k+1 , i = 1, 2, . . . , m,
r = 1, 2, . . . , l, j = 1, 2, . . . , n, k = 1, 2, . . . .
Definition 1.1. By a solution of problems (1.1), (1.4) ((1.5)) with initial condition,
we mean that any function u(t, x) for which the following conditions are valid:
(1) If −δ ≤ t ≤ 0, then u(t, x) = Φ(t, x), ∂u(t,x)
= Ψ(t, x).
∂t
(2) If 0 ≤ t ≤ t̄1 = t1 , then u(t, x) coincides with the solution of the problems
(1.1)–(1.3) and (1.4) ((1.5)) with initial condition.
(3) If t̄k < t ≤ t̄k+1 , t̄k ∈ {tk } \ ({tki } ∪ {tkr } ∪ {tkj }), then u(t, x) coincides
with the solution of the problems (1.1)–(1.3) and (1.4) ((1.5)).
(4) If t̄k < t ≤ t̄k+1 , t̄k ∈ {tki } ∪ {tkr } ∪ {tkj }, then u(t, x) satisfies (1.4) ((1.5))
and coincides with the solution of the problem
m
i
X
∂2 h +
+
g
u((t
−
τ
)
,
x)
u(t
,
x)
+
i
i
∂t2
i=1
= a(t)h(u(t+ , x))∆u(t+ , x) − q(t, x)f (u(t+ , x))
+
l
X
ar (t)hr (u((t − σr )+ , x))∆u((t − σr )+ , x)
r=1
−
n
X
qj (t, x)fj (u((t − ρj )+ , x)),
t 6= tk , (t, x) ∈ R+ × Ω = G,
j=1
u(t̄+
k , x) = u(t̄k , x),
ut (t̄+
k , x) = ut (t̄k , x),
fort̄k ∈ ({tki } ∪ {tkr } ∪ {tkj }) \ {tk },
or
u(t̄+
k , x) = hki (t̄k , x, u(t̄k , x)),
ut (t̄+
k , x) = pki (t̄k , x, ut (t̄k , x)),
for t̄k ∈ ({tki } ∪ {tkr } ∪ {tkj }) ∩ {tk },
where the number ki is determined by the equality t̄k = tki .
We introduce the notation:
Γk = {(t, x) : t ∈ (tk , tk+1 ), x ∈ Ω},
Γ = ∪∞
k=0 Γk ,
Γ̄k = {(t, x) : t ∈ (tk , tk+1 ), x ∈ Ω̄}, Γ̄ = ∪∞
k=0 Γ̄k ,
Z
v(t) =
u(t, x) dx, q(t) = min q(t, x), qj (t) = min qj (t, x).
Ω
x∈Ω̄
x∈Ω̄
Definition 1.2. The solution u ∈ C 2 (Γ) ∩ C 1 (Γ̄) of problems (1.1), (1.4) ((1.5)) is
called non-oscillatory in the domain G if it is either eventually positive or eventually
negative. Otherwise, it is called oscillatory.
2. Oscillation properties for (1.1), (1.4)
For the main result of this article, we need following lemmas.
4
J. C. YANG, A. P. LIU, G. J. LIU
EJDE-2013/27
Lemma 2.1. Let u ∈ C 2 (Γ) ∩ C 1 (Γ̄) be a positive solution of (1.1), (1.4) in G,
then function w(t) satisfies the impulsive differential inequality
m
n
m
X
X
X
w00 (t) + C 1 −
gi q(t)w(t) +
Cj 1 −
gi qj (t)w(t − ρj ) ≤ 0,
i=1
j=1
t 6= tk ,
i=1
(2.1)
ak ≤
bk ≤
where w(t) = v(t) +
w(t+
k)
≤ ak ,
k = 1, 2, . . . ,
(2.2)
w0 (t+
k)
≤ bk ,
w0 (tk )
k = 1, 2, . . . ,
(2.3)
w(tk )
Pm
i=1 gi v(t
− τi ).
Proof. Let u(t, x) be a positive solution of the problem (1.1), (1.4) in G. Without
loss of generality, we may assume that there exists a T > 0, t0 > T such that
u(t, x) > 0, u(t − τi , x) > 0, i = 1, 2, . . . , m, u(t − σr , x) > 0, r = 1, 2, . . . , l,
u(t − ρj , x) > 0, j = 1, 2, . . . n, for any (t, x) ∈ [t0 , ∞) × Ω.
For t ≥ t0 , t 6= tk , k = 1, 2, . . . , integrating (1.1) with respect to x over Ω yields
Z
Z
m
i
X
d2 h
u(t,
x)
dx
+
g
u(t − τi , x) dx
i
2
dt
Ω
Ω
i=1
Z
Z
= a(t)
h(u)∆u dx −
q(t, x)f (u(t, x)) dx
Ω
+
−
Ω
l
X
Z
ar (t)
r=1
n Z
X
hr (u(t − σr , x))∆u(t − σr , x) dx
Ω
qj (t, x)fj (u(t − ρj , x)) dx.
j=1
Ω
By Green’s formula and the boundary condition, we have
Z
Z
Z
∂u
h(u)∆u dx =
h(u) ds −
h0 (u)|gradu|2 dx
∂n
Ω
∂Ω
Ω
Z
=−
h0 (u)|gradu|2 dx ≤ 0,
Ω
Z
hr (u(t − σr , x))∆u(t − σr , x) dx ≤ 0.
Ω
From condition (H2), we can easily obtain
Z
Z
q(t, x)f (u(t, x)) dx ≥ Cq(t)
u(t, x) dx,
Ω
Ω
Z
Z
qj (t, x)fj (u(t − ρj , x)) dx ≥ Cj qj (t)
u(t − ρj , x) dx.
Ω
Ω
From the above it follows that
m
n
i
X
X
d2 h
v(t)
+
g
v(t
−
τ
)
+
Cq(t)v(t)
+
Cj qj (t)v(t − ρj ) ≤ 0.
i
i
dt2
i=1
j=1
EJDE-2013/27
OSCILLATION FOR HYPERBOLIC EQUATIONS
Set w(t) = v(t) +
Pm
i=1 gi v(t
5
− τi ), we have w(t) ≥ v(t), then we can obtain
w00 (t) + Cq(t)v(t) +
n
X
Cj qj (t)v(t − ρj ) ≤ 0,
t 6= tk .
j=1
From this inequality, we have w00 (t) < 0, and w0 (t) > 0, w(t) > 0 for t ≥ t0 . In
fact, if w0 (t) < 0, there exists t1 ≥ t0 such that w0 (t1 ) < 0. Hence we have
Z t
Z t
0
w(t) − w(t1 ) =
w (s)ds ≤
w0 (t1 )ds = w0 (t1 )(t − t1 ),
t1
t1
lim w(t) = −∞.
t→+∞
This is a contradiction, so w0 (t) > 0. Because
m
X
v(t) = w(t) −
gi v(t − τi )
i=1
m
m
h
i
X
X
= w(t) −
gi w(t − τi ) −
gi v(t − 2τi )
i=1
= w(t) −
i=1
m
X
m
m
i
hX
X
gi
gi v(t − 2τi )
gi w(t − τi ) +
i=1
i=1
i=1
m
X
gi w(t),
≥ 1−
i=1
m
X
v(t − ρj ) ≥ 1 −
gi w(t − ρj ).
i=1
Hence, we obtain
m
n
m
X
X
X
w00 (t) + C 1 −
gi q(t)w(t) +
Cj 1 −
gi qj (t)w(t − ρj ) ≤ 0,
i=1
j=1
t 6= tk .
i=1
For t ≥ t0 , t = tk , k = 1, 2, . . . , from (1.2), (1.3) and condition (H4), we obtain
According to the v(t) =
R
Ω
ak ≤
ak ≤
u(t+
k , x)
≤ ak ,
u(tk , x)
(2.4)
bk ≤
ut (t+
k , x)
≤ bk .
ut (tk , x)
(2.5)
u(t, x) dx, we obtain
v(t+
k)
≤ ak ,
v(tk )
bk ≤
v 0 (t+
k)
≤ bk .
0
v (tk )
Pm
i=1 gi v(t − τi ), we finally have
w(t+
w0 (t+ )
k)
ak ≤
≤ ak , bk ≤ 0 k ≤
w(tk )
w (tk )
Because w(t) = v(t) +
bk .
Hence we obtain that w(t) is a solution of impulsive differential inequality (2.1)–
(2.3). This completes the proof.
Lemma 2.2 ([5, Theorem 1.4.1]). Assume that
(A1) the sequence {tk } satisfies 0 < t0 < t1 < t2 < . . . , limk→∞ tk = ∞;
6
J. C. YANG, A. P. LIU, G. J. LIU
EJDE-2013/27
(A2) m(t) ∈ P C 1 [R+ , R] is left continuous at tk for k = 1, 2, . . . ;
(A3) for k = 1, 2, . . . and t ≥ t0 ,
m0 (t) ≤ p(t)m(t) + q(t), t 6= tk ,
m(t+
k ) ≤ dk m(tk ) + ek ,
where p(t), q(t) ∈ C(R+ , R), dk ≥ 0 and ek are constants. P C denote the
class of piecewise continuous function from R+ to R, with discontinuities
of the first kind only at t = tk , k = 1, 2, . . . .
Then
Y
m(t) ≤ m(t0 )
dk exp
Z
+
Z
p(s)ds +
t0
t0 <tk <t
X
t
Y
dj exp
t0 <tk <t tk <tj <t
t
Y
dk exp
Z
t0 s<tk <t
Z
t
t
p(r)dr q(s)ds
s
p(s)ds ek .
tk
Lemma 2.3 ([11]). Let w(t) be an eventually positive (negative) solution of the
differential inequality (2.1)–(2.3). Assume that there exists T ≥ t0 such that w(t) >
0 (w(t) < 0) for t ≥ T . If
Z t Y
bk
ds = +∞
(2.6)
lim
t→+∞ t
ak
0 t0 <tk <s
hold, then w0 (t) ≥ 0 (w0 (t) ≤ 0) for t ∈ [T, tl ] ∪ ∪+∞
k=l (tk , tk+1 ] , where l = min{k :
tk ≥ T }.
The following theorem is the first main result of this article.
Theorem 2.4. If condition (2.6) and the following condition holds,
Z t Y
ak
F (s)ds = +∞,
lim
t→+∞ t
0 t0 <tk <s bk
(2.7)
where
m
n
m
X
X
X
F (s) = C 1 −
gi q(s) +
Cj 1 −
gi qj (s) exp(−δz(t0 )).
i=1
j=1
i=1
Then each solution of (1.1)–(1.3), (1.4) oscillates in G.
Proof. Let u(t, x) be a non-oscillatory solution of (1.1), (1.4). Without loss of
generality, we can assume that there exists T > 0, t0 ≥ T , such that u(t, x) > 0,
u(t − τi , x) > 0, i = 1, 2, . . . , m, u(t − σr , x) > 0, r = 1, 2, . . . , l, u(t − ρj , x) > 0,
j = 1, 2, . . . , n for any (t, x) ∈ [t0 , ∞) × Ω. From Lemma 2.1, we know that w(t) is
a solution of (2.1)–(2.3).
For t ≥ t0 , t 6= tk , k = 1, 2, . . . , define
z(t) =
w0 (t)
,
w(t)
t ≥ t0 .
(2.8)
From Lemma 2.3, we have z(t) ≥ 0, t ≥ t0 , w0 (t) − z(t)w(t) = 0. We may assume
that w(t0 ) = 1, thus in view of (2.1)–(2.3) we have that for t ≥ t0 ,
Z t
w(t) = exp
z(s)ds ,
(2.9)
t0
EJDE-2013/27
OSCILLATION FOR HYPERBOLIC EQUATIONS
w0 (t) = z(t) exp
t
Z
z(s)ds ,
7
(2.10)
t0
00
2
w (t) = z (t) exp
Z
t
Z t
0
z(s)ds + z (t) exp
z(s)ds .
t0
(2.11)
t0
We substitute (2.9)–(2.11) into (2.1) and obtain
Z t
Z t
2
0
z (t) exp
z(s)ds + z (t) exp
z(s)ds
t0
t0
m
Z t
X
+C 1−
gi q(t) exp
z(s)ds
t0
i=1
+
n
X
Cj
m
Z
X
1−
gi qi (t) exp
j=1
t−ρj
z(s)ds ≤ 0.
t0
i=1
Hence we have
m
X
z 2 (t) + z 0 (t) + C 1 −
gi q(t)
i=1
+
n
X
Cj
m
Z
X
1−
gi qj (t) exp −
j=1
t
z(s)ds ≤ 0,
t 6= tk ,
z(s)ds ≤ 0,
t 6= tk .
t−ρj
i=1
then we have
m
X
gi q(t)
z 0 (t) + C 1 −
i=1
+
n
X
Cj
m
Z
X
1−
gi qj (t) exp −
j=1
t
t−ρj
i=1
From above inequality and condition bk ≤ ak , we know that z(t) is non-increasing,
then z(t) ≤ z(t0 ), for t ≥ t0 , we obtain
m
n
m
X
X
X
z 0 (t) + C 1 −
gi q(t) +
Cj 1 −
gi qj (t) exp(−δz(t0 )) ≤ 0, t 6= tk .
i=1
j=1
i=1
From (2.2), (2.3) and (2.8), we obtain
z(t+
k)=
w0 (t+
bk w0 (tk )
bk
k)
=
z(tk ),
+ ≤
ak w(tk )
ak
w(tk )
so we can easily obtain
m
n
m
X
X
X
z 0 (t) ≤ −C 1 −
gi q(t) −
Cj 1 −
gi qj (t) exp(−δz(t0 )) t 6= tk ,
i=1
j=1
i=1
bk
z(t+
z(tk ).
k)≤
ak
Let
m
n
m
X
X
X
−F (t) = −C 1 −
gi q(t) −
Cj 1 −
gi qj (t) exp(−δz(t0 )).
i=1
j=1
i=1
8
J. C. YANG, A. P. LIU, G. J. LIU
EJDE-2013/27
Then according to Lemma 2.2, we have
Y bk Z t Y bk
z(t) ≤ z(t0 )
+
(−F (s)) ds
a
t0 s<tk <t ak
t0 <tk <t k
Z t Y
i
Y bk h
ak
z(t0 ) −
F (s)ds < 0.
=
a
t0 t0 <tk <s bk
t0 <tk <t k
Since z(t) ≥ 0, this is a contradiction. The proof is complete.
3. Oscillation properties of the problem (1.1), (1.5)
For the second main theorem, we need following lemma.
Lemma 3.1. Let u ∈ C 2 (Γ) ∩ C 1 (Γ̄) be a positive solution of (1.1), (1.5) in G,
then function w(t) satisfies the impulsive differential inequality
m
n
m
X
X
X
w00 (t) + C 1 −
gi q(t)w(t) +
Cj 1 −
gi qj (t)w(t − ρj ) ≤ 0, t 6= tk ,
i=1
j=1
i=1
(3.1)
ak ≤
bk ≤
where w(t) = v(t) +
w(t+
k)
≤ ak ,
k = 1, 2, . . . ,
(3.2)
w0 (t+
k)
≤ bk ,
0
w (tk )
k = 1, 2, . . . ,
(3.3)
w(tk )
Pm
i=1 gi v(t
− τi ).
Proof. Let u(t, x) be a positive solution of the problem (1.1), (1.5) in G. Without
loss of generality, we may assume that there exists a T > 0, t0 > T such that
u(t, x) > 0, u(t − τi , x) > 0, i = 1, 2, . . . , m, u(t − σr , x) > 0, r = 1, 2, . . . , l,
u(t − ρj , x) > 0, j = 1, 2, . . . n, for any (t, x) ∈ [t0 , ∞) × Ω. For t ≥ t0 , t 6= tk ,
k = 1, 2, . . . , integrating (1.1) with respect to x over Ω yields
Z
Z
m
i
X
d2 h
g
u(t
−
τ
,
x)
dx
u(t,
x)
dx
+
i
i
dt2 Ω
Ω
i=1
Z
Z
= a(t)
h(u)∆u dx −
q(t, x)f (u(t, x)) dx
Ω
+
−
Ω
l
X
Z
ar (t)
r=1
n Z
X
hr (u(t − σr , x))∆u(t − σr , x) dx
Ω
qj (t, x)fj (u(t − ρj , x)) dx.
j=1
Ω
By Green’s formula and the boundary condition, we have
Z
Z
Z
∂u
h(u)∆u dx =
h(u) ds −
h0 (u)|gradu|2 dx
∂n
Ω
∂Ω
Ω
Z
Z
=−
h(u)ϕ(t, x)u ds −
h0 (u)|gradu|2 dx
∂Ω
Ω
Z
≤−
h0 (u)|gradu|2 dx ≤ 0,
Ω
EJDE-2013/27
OSCILLATION FOR HYPERBOLIC EQUATIONS
9
Z
hr (u(t − σr , x))∆u(t − σr , x) dx ≤ 0.
Ω
The rest of the proof is similar to the one in Lemma 2.1. We omit it.
The following theorem is the second main result of this article.
Theorem 3.2. If conditions (2.6) and (2.7) hold, then each solution of (1.1)–(1.3),
(1.5) oscillates in G.
The proof of the above theorem is similar to that of Theorem 2.4. We omit it.
4. Examples
Example 4.1. Consider the equation
i
2
1
π
π
π
∂2 h
u(t,
x)
+
u(t
−
,
x)
= u2 ∆u − ueu + et u2 (t − , x)∆u(t − , x)
2
∂t
2
2
2
2
3π
u2 (t− 3π
2
t
,x)
2
, x)e
− (x + 1)e u(t −
,
2
t > 1, t 6= 2k , (t, x) ∈ R+ × Ω = G,
u((2k )+ , x) = (4 + sin 2k cos x)u(2k , x),
ut ((2k )+ , x) = (2 + sin 2k cos x)ut (2k , x),
k = 1, 2, . . . ,
k = 1, 2, . . . ,
with the boundary condition
u = 0,
(t, x) ∈ R+ × ∂Ω,
2
2
where a(t) = 1, a1 (t) = et , τ1 = π2 , σ1 = π2 , ρ1 = 3π
2 , h(u) = u , h1 (u) = u ,
1
u2
u2
2
t
k
f (u) = ue , f1 (u) = ue , q(t, x) = 1, q1 (t, x) = (x + 1)e , g1 = 2 , tk = 2 . It is
easy to verify that the condition (H1)–(H4) and the conditions of Theorem 2.4 are
satisfied. Hence the all solutions of above problem oscillate.
Example 4.2. Consider the equation
i
∂2 h
1
π
u(t, x) + u(t − , x)
2
∂t
2
2
2
3π
π
π
3π
2
u2
t 2
= u ∆u − ue + e u (t − , x)∆u(t − , x) − (x2 + 1)et u(t −
, x)eu (t− 2 ,x) ,
2
2
2
t > 1, t 6= 3k, (t, x) ∈ R+ × Ω = G,
u((3k)+ , x) = (4 + sin 3k cos x)u(3k, x),
+
ut ((3k) , x) = (2 + sin 3k cos x)ut (3k, x),
k = 1, 2, . . . ,
k = 1, 2, . . . ,
with the boundary condition
∂u
+ t2 x2 u = 0, (t, x) ∈ R+ × ∂Ω,
∂n
2
2
where a(t) = 1, a1 (t) = et , τ1 = π2 , σ1 = π2 , ρ1 = 3π
2 , h(u) = u , h1 (u) = u ,
1
u2
u2
2
t
f (u) = ue , f1 (u) = ue , q(t, x) = 1, q1 (t, x) = (x + 1)e , g1 = 2 , tk = 3k,
ϕ(t, x) = t2 x2 . It is easy to verify that the condition H) and condition of Theorem
3.2 are satisfied. Hence the all solutions of the above problem oscillate.
Acknowledgments. This research was partially supported by grants from the National Basic Research Program of China, nos. 2010CB950904 and 2011CB710604.
10
J. C. YANG, A. P. LIU, G. J. LIU
EJDE-2013/27
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Jichen Yang
School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei,
430074, China
E-mail address:
yjch 0102@hotmail.com
Anping Liu (corresponding author)
School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei,
430074, China
E-mail address: wh apliu@sina.com
Guangjie Liu
School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei,
430074, China
E-mail address: guangjieliu@foxmail.com