Electronic Journal of Differential Equations, Vol. 2004(2004), No. 24, pp. 1–6.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
OSCILLATION OF NONLINEAR IMPULSIVE HYPERBOLIC
EQUATIONS WITH SEVERAL DELAYS
ANPING LIU, LI XIAO, & TING LIU
Abstract. In this paper, we study oscillatory properties of solutions to nonlinear impulsive hyperbolic equations with several delays. Sufficient conditions
for oscillations of the solutions are established
1. Introduction
The theory of partial functional differential equations can be applied to fields,
such as to biology, population growth, engineering, medicine, physics and chemistry.
In the last few years, a few of papers have been published on oscillation theory
of partial differential equations. The qualitative theory of this class of equations,
however, is still in an initial stage of development. We may easily visualize situations
in nature where abrupt changes such as shock and disasters may occur. These
phenomena are short-time perturbations. Consequently, it is natural to assume,
in modelling these problems, that these perturbations act instantaneously, that is,
in the form of impulses. In 1991, the first paper [8] on this class of equations was
published. But for instance, on oscillation theory of impulsive partial differential
equations only a few of papers have been published. Recently, Bainov, Minchev, Fu
and Luo [4, 5, 9, 10, 19] 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 with
several delays. In this paper, we discuss the oscillatory properties of solutions
for nonlinear impulsive hyperbolic equations with several delays (1.1), under the
boundary condition (1.4). It should be noted that the equation we discuss here is
nonlinear and that we could not find work for oscillations of this kind of problem.
m
X
∂2u
=
a(t)h(u)∆u
+
ai (t)hi (u(t − τi , x))∆u(t − τi , x)
∂t2
i=1
− q(t, x)f (u(t, x)) −
n
X
gj (t, x)fj (u(t − σj , x)),
(1.1)
j=1
t 6= tk ,
(t, x) ∈ R+ × Ω = G,
2000 Mathematics Subject Classification. 35L10, 35R12, 35R10.
Key words and phrases. Impulse, hyperbolic differential equation, oscillation, delay.
c
2004
Texas State University - San Marcos.
Submitted December 1, 2003. Published February 19, 2004.
Supported by grant 10071026 from the National Natural Science Foundation of China, and
by grant 40373003 from the Natural Science Foundation of China.
1
2
ANPING LIU, LI XIAO, & TING LIU
EJDE-2004/24
−
u(t+
k , x) − u(tk , x) = qk u(tk , x),
ut (t+
k , x)
−
ut (t−
k , x)
= bk ut (tk , x),
(1.2)
k = 1, 2, . . . .
(1.3)
with the boundary condition
∂u
= 0,
∂n
(t, x) ∈ R+ × ∂Ω,
(1.4)
and the initial condition u(t, x) = Φ(t, x), (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 , σj }, Φ(t, x) ∈ C 2 ([−δ, 0] × Ω, R).
Assume that the following conditions are fulfilled:
(H1) a(t), ai (t) ∈ P C(R+ , R+ ), τi = const. > 0, σj = const. > 0, i = 1, 2, . . . m,
j = 1, 2, . . . n, q(t, x), gj (t, x) ∈ C(R+ × Ω, (0, ∞)); 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) h0 (u), h0i (u), f (u), fj (u) ∈ C(R, R); f (u)/u ≥ C = const. > 0, fj (u)/u ≥
Cj = const. > 0, for u 6= 0; qk > −1, bk > −1, bk < qk , 0 < t1 < t2 < · · · <
tk < . . . , limt→∞ 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, . . . .
Let us construct the sequence {t̄k } = {tk } ∪ {tkτi } ∪ {tkσj }, where tkτi = tk +
τi , tkσj = tk + σj and t̄k < t̄k+1 , i = 1, 2, . . . , m, k = 1, 2, . . . .
Definition. By a solution of problem (1.1), (1.4) 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).
(2) If 0 ≤ t ≤ t̄1 = t1 , then u(t, x) coincides with the solution of the problem
(1.1)–(1.4) with initial condition. S
(3) If t̄k < t ≤ t̄k+1 , t̄k ∈ {tk } \ ({tki } {tkj }), then u(t, x) coincides with the
solution of the problem (1.1)–(1.4).
S
(4) If t̄k < t ≤ t̄k+1 , t̄k ∈ {tki } {tkj }, then u(t, x) coincides with the solution
of the problem (1.4) and the following equations
m
X
∂2u
+
+
=
a(t)h(u(t
,
x))∆u(t
,
x)
+
ai (t)hi (u((t − τi )+ , x))∆u((t − τi )+ , x)
∂t2
i=1
−q(t, x)f (u(t+ , x)) −
n
X
gj (t, x)fj (u((t − σj+ ), x))
(t, x) ∈ R+ × Ω = G
j=1
u(t̄+
k , x) = u(t̄k , x),
ut (t̄+
k , x) = ut (t̄k , x),
for t̄k ∈ ({tkτi } ∪ {tkσj }) \ {tk },
or
u(t̄+
k , x) = (1 + qki )u(t̄k , x),
ut (t̄+
k , x) = (1 + bki )ut (t̄k , x)
for t̄k ∈ ({tkτi } ∪ {tkσj }) ∩ {tk }.
Where the number ki is determined by the equality t̄k = tki .
EJDE-2004/24
OSCILLATION OF NONLINEAR EQUATIONS
3
S∞
We introduce the notation: Γk = {(t, x)
: t ∈ (tk , tk+1 ),Rx ∈ Ω}, Γ = k=0 Γk ,
S∞
Γ̄k = {(t, x) : t ∈ (tk , tk+1 ), x ∈ Ω̄}, Γ̄ = k=0 Γ̄k , v(t) = Ω u(t, x)dx and p(t) =
minq(t, x), pj (t) = mingj (t, x), x ∈ Ω̄.
Definition.The solution u ∈ C 2 (Γ)∩C 1 (Γ̄) of problem (1.1), (1.4) is called nonoscillatory in the domain G if it is either eventually positive or eventually negative.
Otherwise, it is called oscillatory.
2. Oscillation properties of the problem (1.1), (1.4)
For the main theorem of this paper, we need following lemmas.
Lemma 2.1. Let u ∈ C 2 (Γ) ∩ C 1 (Γ̄) be a positive solution of (1.1), (1.4) in G,
then function v(t) satisfies the impulsive differential inequality
v 00 (t) + Cp(t)v(t) +
n
X
Cj pj (t)v(t − σj ) ≤ 0, , t 6= tk ,
(2.1)
j=1
v(t+
k ) = (1 + qk )v(tk ) k = 1, 2, . . . .
v
0
(t+
k)
0
= (1 + bk )v (tk ) k = 1, 2, . . . .
(2.2)
(2.3)
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 u(t, x) > 0, u(t − τi , x) > 0, i =
1, 2, . . . , m, u(t − σj , 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
Z
m
X
d2
[
udx]
=
a(t)
h(u)∆udx
+
a
(t)
hi (u(t − τi , x))∆u(t − τi , x)dx
i
dt2 Ω
Ω
Ω
i=1
Z
n Z
X
−
q(t, x)f (u(t, x))dx −
gj (t, x)fj (u(t − σj , x)).
Ω
j=1
Ω
(2.4)
By Green’s formula and the boundary condition, we have
Z
Z
Z
Z
∂u
h(u)∆udx =
h(u) ds −
h0 (u)|gradu|2 dx ≤ −
h0 (u)|gradu|2 dx ≤ 0,
∂n
Ω
∂Ω
Ω
Ω
Z
hi (u(t − τi , x))∆u(t − τi , x)dx ≤ 0.
Ω
From condition (H2), we can easily obtain
Z
Z
q(t, x)f (u(t, x))dx ≥ Cp(t)
u(t, x)dx,
Ω
Ω
Z
Z
qj (t, x)fj (u(t − σj , x))dx ≥ Cj pj (t)
u(t − σj , x)dx.
Ω
Ω
It follows that from above that
n
X
v 00 + Cp(t)v(t) +
Cj pj (t)v(t − σj ) ≤ 0,
(t ≥ t0 , t 6= tk ).
j=1
Where v(t) > 0. For t > t0 , t = tk , k = 1, 2, . . . , we have
Z
Z
Z
+
−
u(tk , x)dx −
u(tk , x)dx = qk
u(tk , x)dx,
Ω
Ω
Ω
(2.5)
4
ANPING LIU, LI XIAO, & TING LIU
Z
Ω
ut (t+
k , x)dx −
Z
ut (t−
k , x)dx = bk
Ω
EJDE-2004/24
Z
ut (tk , x)dx.
Ω
This implies
v
0
(t+
k)
v(t+
k ) = (1 + qk )v(tk ),
(2.6)
0
(2.7)
= (1 + bk )v (tk ) k = 1, 2, . . . .
Hence we obtain that v(t) > 0 is a positive solution of differential inequality (2.1)–
(2.3). This completes the proof.
Definition. The solution v(t) of differential inequality (2.1)–(2.3) is called eventually positive (negative) if there exists a number t∗ such that v(t) > 0(v(t) < 0)
for t ≥ t∗ .
Lemma 2.2 ([2, Theorem 1.4.1]). Assume that
(i) m(t) ∈ P C 1 [R+ , R] is left continuous at tk for k = 1, 2, . . . ,
(ii) For k = 1, 2, . . . and t ≥ t0 ,
m0 (t) ≤ p(t)m(t) + q(t)
m(t+
k)
(t 6= tk ),
≤ dk m(tk ) + ek .
where p(t), q(t) ∈ C(R+ , R), dk ≥ 0 and ek are real constants, P C 1 [R+ , R] = {x :
R+ → R; x(t) is continuous and continuously differentiable everywhere except some
−
−
0 +
0 −
0
0 −
tk at which x(t+
k ), x(tk ), x (tk ) and x (tk ) exist and x(tk ) = x(tk ), x (tk ) = x (tk )}.
Then
Z t
Z t Y
Z t
Y
m(t) ≤ m(t0 )
dk exp(
p(s)ds) +
dk exp(
p(r)dr)q(s)ds
t0
t0 <tk <t
+
X
Y
t0 s<tk <t
s
Z t
dj exp(
p(s)ds)ek .
tk
t0 <tk <t tk <tj <t
From the above lemma, we can obtain the following result; see also [19].
Lemma 2.3. Let v(t) be eventually positive (negative) solution of differential inequality (2.1)–(2.3). Assume that there exists T ≥ t0 such that v(t) > 0(v(t) < 0)
for t ≥ T . If the following condition holds,
Z t Y
1 + bk
ds = +∞,
(2.8)
lim
t→+∞ t
1
+ qk
0 t0 <tk <s
S S+∞
then v 0 (t) ≥ 0(v 0 (t) ≤ 0) for t ∈ [T, tl ] ( k=l (tk , tk+1 ]), where l = min{k : tk ≥
T }.
Theorem 2.4. If condition (2.8) hold and, for some j0 ,
Z t Y
1 + qk
lim
pj (s)ds = +∞,
t→+∞ t
1 + bk 0
0 t <t <s
0
(2.9)
k
then each solution of (1.1)–(1.4) oscillates in G.
Proof. Let u(t, x) be a nonoscillatory solution of (1.1), (1.4). Without loss of generality, we can assume that u(t, x) > 0, u(t−τi , x) > 0, i = 1, 2, . . . , m, u(t−σj , x) > 0,
j = 1, 2, . . . , n for any (t, x) ∈ [t0 , ∞) × Ω. From Lemma 2.1, we know that v(t) is
EJDE-2004/24
OSCILLATION OF NONLINEAR EQUATIONS
5
a positive solution of (2.1)–(2.3). Thus from Lemma 2.3, we can find that v 0 (t) ≥ 0
for t ≥ t0 .
For t ≥ t0 , t 6= tk , k = 1, 2, . . . , define
w(t) =
v 0 (t)
,
v(t)
t ≥ t0 .
Then we have w(t) > 0, t ≥ t0 , v 0 (t)−w(t)v(t) = 0. We may assume that v(t0 ) = 1,
thus in view of (2.1)–(2.3) we have that for t ≥ t0 ,
Z t
v(t) = exp(
w(s)ds)
(2.10)
t0
v 0 (t) = w(t) exp(
Z
t
w(s)ds)
(2.11)
t0
00
2
Z
t
v (t) = w (t) exp(
Z
0
t
w(s)ds) + w (t) exp(
t0
w(s)ds)
We substitute (2.10)–(2.12) into (2.1) and can obtain
Z t
Z t
Z
2
0
w (t) exp(
w(s)ds) + w (t) exp(
w(s)ds) + Cj0 pj0 (t)exp(
t0
(2.12)
t0
t0
t−σj0
w(s)ds) ≤ 0.
t0
Hence, we have
w2 (t) + w0 (t) + Cj0 pj0 (t)exp(−
Z
t
w(s)ds) ≤ 0 .
t−σj0
From above and condition bk < qk , it is easy to see that the function w(t) is
non-increasing for t ≥ t1 ≥ δ + t0 . Thus w(t) ≤ w(t1 ), for t ≥ t1 , which implies
w0 (t) + Cj0 pj0 (t) exp(−δw(t1 )ds) ≤ 0,
t ≥ t1 .
From (2.2), (2.3) we get
w(t+
k)=
v 0 (t+
1 + bk
k)
w(tk ),
+ =
1
+ qk
v(tk )
k = 1, 2, . . . .
It follows that
w0 (t) ≤ −Cj0 pj0 (t)exp(−δw(t1 )ds)
(t 6= tk ),
(2.13)
1 + bk
w(t+
w(tk ) (t = tk ).
(2.14)
k)=
1 + qk
Using Lemma 2.2, we obtain
Y 1 + bk Z t Y 1 + b k
w(t) ≤ w(t0 )
+
(−Cj0 pj0 (s) exp(−δw(t1 )))ds
1 + qk
t0 s<tk <t 1 + qk
t0 <tk <t
Z t Y
Y 1 + bk
1 + qk
=
{w(t0 ) −
Cj0 pj0 (s) exp(−δw(t1 ))ds}.
1
+
q
k
t0 t <t <s 1 + bk
t <t <t
0
k
0
k
Since w(t) > 0, the last inequality contradicts (2.9). Therefore, the proof is complete.
It should be noted that obviously all solutions of problem (1.1)–(1.2) are oscillatory if there exists a subsequence nk of n such that qnk < −1, for k = 1, 2, . . . .
So we discuss only the case of qk > −1.
6
ANPING LIU, LI XIAO, & TING LIU
EJDE-2004/24
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Department of Mathematics and Physics, China University of Geosciences, Wuhan,
Hubei, 430074, China
E-mail address, Anping Liu: wh apliu@263.net