Journal
of Applied Mathematics
and Stochastic Analysis, 11:2
(1998), 193-208.
OSCILLATION CRITERIA FOR HIGH ORDER DELAY
PARTIAL DIFFERENTIAL EQUATIONS 1
XINZHI LIU and XILIN FU 2
University
of Waterloo, Department of Applied Mathematics
Waterloo, Ontario, Canada N2L 3G1
(Received October, 1996; Revised January, 1997)
This paper studies a class of high order delay partial differential equations.
Employing high order delay differential inequalities, several oscillation criteria are established for such equations subject to two different boundary
conditions. Two examples are also given.
Key words: Oscillation, Higher Order Delay Partial Differential Equations, Differential Inequality, Eventual Positive Solutions.
AMS subject classifications: 35B05. 34L40.
1. Introduction
The oscillation theory of delay differential equations has been studied by numerous
authors and the number of papers published in this area is enormous. For an excellent exposition of the basic theory, see [5]. In recent years, there has been an increasing interest in oscillation theory of delay partial differential equations, see [6-10]
and references therein. However, the corresponding theory is still in its initial stage
of development. In this paper, we shall investigate a class of high order delay partial
differential equations which will be described in Section 2. In Section 3, we shall establish several oscillation criteria for high order delay partial differential equations subject to two kinds of boundary conditions, employing Green’s theorem and high order
delay differential inequalities. We then develop, in Section 4, some results on eventual positive and eventual negative solutions of high order differential inequalities,
which enable us, in addition to their independent interests, to obtain in Section 5,
further oscillation criteria for high order delay partial differential equations. To illustrate our results, two examples are also given.
1Research supported by NSERC-Canada.
2On leave from Shandong Normal University, Jinan, Shandong 250014, PR
China.
Printed in the U.S.A.
()1998 by North Atlantic
Science Publishing Company
193
XINZHI LIU and XILIN FU
194
2. Prehminaries
We shall consider the following nonlinear high order delay partial differential
equation
ff-m[u + A(t)u(x, v)] + p(x, t)u + q(x, t)f(u(x, t
= a(t)Au + E aj(t)Au(x, aj(t)),(x,t)e x R+
r))
G,
(2.1)
j=l
where m is an even positive integer, > 0 and a > 0 are constants. Let fl be a
bounded domain in R n with piecewise boundary OF/, A is the Laplacian in Rn;
p,q C[R + x ,R + ],
A E cm[R + ,R];
j 1, 2,...,
a, aj C[R + ,R + ],
t
and lim r(t)
in
is
<
R
nondecreasing
t,
R],
+ cx3,
rj(t)
f C[R,
rj C[R +, +
t--. +
j 1, 2,...,
We shall consider two kinds of boundary conditions
"
.
;
Ou
ON t- 7(x’ )u
and
0,
o, (, ) e O R +
(, ) e 0e a +,
(B1)
(B)
where N is the unit exterior normal vector to 0, 7(x, t) is a nonnegative continuous
function on c3 x R +.
Definition 2.1: The solution u(x,t) of system (2.1) satisfying certain boundary
conditions is called oscillatory in the domain G if for each positive number #, there
exists a point (Xo, to) 2 x lit, + cx) such that U(Xo, to) O.
3. Oscillation Criteria
we shall establish oscillation criteria for problem (2.1) with boundary
condition (B1) and (B2) separately. The basic idea of our approach is to reduce the
study of high order delay partial differential equations to that of high order delay
differential inequalities.
Theorem 3.1: Assume that the following condition (H) holds.
(H) f(u) is convex in R + and f( u) = f(u) < O, u E R +
If the high order delay differential inequalities
In this section
d,
d--t-[U(t + (t)U(t
r)] + P(t)U(t) + Q(t)f(U(t o’)) _< 0
has no eventually positive solutions, then all solutions
are oscillatory in G, where
P(t)
min_ p(x, t),
Q(t)
of the problem (2.1)
under
(B1)
min_ q(x, t).
xEfl
be a nonoscillatory solution of the problem
xEfl
(3.1)
Proof: Let u(x,t)
(2.1) under (B1).
We may assume that u(x,t)> 0 for (x,t)E x[it, +x), where it is a positive
number t o
> it,
such that
(, t -) > 0, (, t ) > 0
and
u(x,rj(t)) > O, (x,t) e ]x[t o, +oo),
j- 1,2,...,g.
Oscillation Criteria
Integrating both sides of system
dm
(2.1)
+
J
u(x
r)dx +
t
_
p(x, t)u(x t)dx
q(x, t)f(u(x, t r))dx
a(t)/Au(x, t)dx + E aj(t)J Au(x, j(t))dx,
a
j=l
t
195
with respect to x over the domain f2, we obtain
u(x, t)dx + a(t)
dt m
for High Order Delay PDEs
t
t 0.
Ft
From Green’s Theorem, it follows that
(3.3)
and
/
_
/
<_ o,
-/7(,j(t))(,y(t))d8
o
j- 1,2,...,t, t
where dS is the surface integral element on 0ft. Since
using Jensen’s inequality, we have
f (u(x, t )dx
where
a]
_
f dx.
Combining
all [o1
(3.2)-(3.5)
+ P(t) u(x,t)dx + Q(t)f
a(t)
/ 7(x, t)u(x, t)dS
if(u)
>_ to,
is convex in
u(x, t )dx
(3.4)
R +, then
(3.)
yields
a
E.aj(t) / 7(x, j(t))u(x, aj(t))dS
<_O, t>_t o.
Thus, we see that the function
u(t)
1
al
/ u(x t)dx
(3.6)
XINZHI LIU and XILIN FU
196
is a positive solution of the inequality
of the theorem.
(3.1) for t _> to,
< 0 for (x, t)
[#, + c), then set
(., t)
(., t), (., t)
Note that since f(-u)= -f(u), u E (0, + c), it
positive solution of the problem (2.1) under (B1)
which contradicts the condition
If u(x, t)
[, + ).
is easy to check that (x,t)is a
which is impossible. This com-
pletes the proof of Theorem 3.1.
The following fact will be used in the proof of Theorem 3.2. Consider the Dirichlet problem
in
Au+u- 0
,
ulo-O,
,-
where
constant. It is well known that the smallest eigenvalue 0 and the corresponding eigenfunction (x) are positive.
Theorem 3.2: Assume that the condition (H) holds. If the high order delay
differential inequality
dm
dt m
[V(t) + A(t)V(t v)] + (A0a(t) + P(t))V(t) + Q(t)f(V(t r)) < 0
(3.7)
has no eventually positive solutions, then all solutions of the problem (2.1) under
are oscillatory in G.
Proof: Let u(x,t) be a solution of the problem (2.1) under (B2) having no zeros
in the domain fix[#, +c), for some #>0. Ifu(x,t)>0 for (x,t) EFt[#,+cx3),
then there exists a t o > tt such that
u(x, t r) > O, u(x, t r) > 0 and u(x, aj(t)) > 0, (x, t) flx [to, +
j
1,2,...,t.
Multiplying both sides of (2.1) by the eigenfunction
to x over the domain fl, we have
(I)(x)
and integrating with respect
d rn
+
I
p(x, t)u(x, t)qP(x)dx +
/ q(x, t)f(u(x,
t
(r))O(x)dx
E aj(t)] Au(x,rj(t))(x)dx, t > t
a(t)J Au(x, t)(x)dx + 3--1
fl
Using Green’s Theorem, we obtain
Au(x, t) ((x)dx
O.
(3.8)
for High Order Delay PDEs
Oscillation Criteria
=
/
197
(3.9)
Au(x, crj(t)). (b(x)dx
o i u(x, rj(t))(b(x)dx,
1, 2,...,
j
.
(3.10)
Using Jensen’s inequality, we have
f(u(x, t cr))O(x)dx
>_
Combining
(x)dx. f
(3.11)
f ((x)dx
(3.8)-(3.10) yields
-
i
Ja(t) u(x, t)((x)dx,
t
>_ to,
i.e., the inequality (3.7) has positive solution
v(t)
f
r ((x)dx ] u(x, t)((x)dx, t >_ to,
J
which contradicts the condition of the theorem.
-u is a positive solution of the
If u(x, t) < 0 for (x, t) E x [#, + oc), then
problem (2.1) under (B2) which also provides a contradiction. The proof of
Theorem 3.2 is complete.
4. High Order Delay Differential Inequalities
From the discussion in Section 3 it follows that the problem of establishing oscillation
criteria for the system (2.1) can be reduced to the investigation of the properties of
the solution of high order delay differential inequalities for the form
XINZHI LIU and XILIN FU
198
m
-m[y(t) + (t)y(t
and
m
d
+ (t)y(t
d--[y(t
7")] + Q(t)f(y(t or)) <_ O, t >_ to,
(4.1)
O, t >_ t o.
(4.2)
7")] + Q(t)f(y(t r))
Along with (4.1) and (4.2), we consider the high order delay differential equation
TM
d
+ (t)y(t
dtm[y(t
o, t >_ to,
’)] + Q(t)f(y(t ))
(4.3)
an even positive integer,
> 0 and r > 0 are constants;
"
fo
om
R
Q
R],
),
C[[,
c),
cm[[to,+
0, +
*o > 0, f e C[R, R]. We h
+
first consider the case A(t) >_ 0.
Assume that y(t) is a nonoscillatory solution of equation (4.3). Let
where
m
is
z(t)
y(t) + A(t)y(t-
We shall use the following lemma.
Lemma 4.1" If z(t) is of definite sign and not identically zero for all sufficiently
large t; there exist a T >_ t o and an integer k, O <_ k <_ m, with rn + k even for
z(t)z(m)(t) >_ O, or rn + k odd for z(t)z(m)(t) <_ O, then
z(t)z(i)(t) > 0 on [r, + c) for 0 < <_ k,
(- 1) i- kz(t)z(i)(t) > 0 on [7", + oc) for k <_ <_ m.
Theorem 4.1- Assume that
0
f(- y)
f(y) for y R +, and that
_< A(t) <_ 1, Q(t) >_ O, t >_ to;
f(Y) > e
y
constant > O, y
(4.4)
(0, + oz)
(4.5)
+ cx,
(4.6)
If
Q(s)[1 A(s cr)]ds
then
the inequality (4.1) has no eventually positive solutions;
the inequality (4.2) has no eventually negative solutions; and
all solutions of the equation (4.3) are oscillatory.
Proof: Let y(t) be an eventually positive solution of the inequality
there exists a t > to, such that
(i)
(ii)
(iii)
(4.1). Then,
y(t) > 0, y(t- r) > 0 and y(t- r) > 0 for all t >_ t 1.
Setting
z(t)
y(t)+ A(t)y(t- 7-), t >_ tl,
we have
z(t) > O, t >_ t 1.
From (4.1), (4.4) and (4.5)it follows that
(4.7)
Oscillation Criteria
z(rn)(t) <_
for High Order Delay PDEs
_
-Q(t)f(y(t- r)) <_ -eQ(t)y(t- or) <_ O, t >_ t 1.
Thus, it follows from Lemma 4.1, that there exists
such that
z()(t) > o, o <_
<_
,
t
an odd number k and a t 2
(-1) i-kz(i)(t)>0, k < <_ m,
z’(t) > O,
Using
(4.5)and (4.7),
z
t1
>_ t
and
It is easy to see that
_
199
(m-1)(t) > 0, t
>_ t 2.
t
(4.8)
t 2.
we have
0
>_ z(m)(t) + Q(t)f(y(t
>_ z(rn)(t) + Q(t) y(t
z(m)(t) + eQ(t)[z(t
r) A(t r)y(t
v
r)], t _> t 2.
r
r)], t >_ t 2.
Note z(t) >_ y(t) for t >_ t2, thus we obtain
0
Since
_
>_ z(m)(t) + eQ(t)[z(t r) A(t r)z(t
z(t) is increasing for t >_ t2,
we have
z(m)(t) + eQ(t)[1
Integrating both sides of
z (m
1)(t)
(4.9) from t 2
z (m
1)(t2)
(t a)]z(t a) _< O,
to
t(t > t2)
ez(t 2 r)
we
J
t
>_ t 2.
(4.9)
get
Q(s)[1 A(s r)]ds.
2
a)(t)
Since z (m> 0 for t _> t2, the above inequality leads to a contradiction in view
of (4.6). This proves assertion (i).
Assertion (it) follows from the fact that if y(t) is an eventually negative solution
of (4.2), then y(t) is an eventually positive solution of (4.1). The proof of the assertion
(iii)
is obvious.
Theorem 4.2: Assume that condition (4.4) holds; f(- y)
f(y) > O, y E R +,
and that f(y) is a monotone increasing function in R +. If for any c > O,
Q(s)f([1 A(s r)]c)ds
+ oc,
then conclusions (i)-(iii) of Theorem 4.1 remain true.
Proof: Let y(t) be an eventually positive solution of inequality
there exists a t I > t o such that
y(t) > 0, y(t-r) > 0 and y(t-r) > 0 for all t _> ta.
(4.10)
(4.1). Then,
XINZHI LIU and XILIN FU
200
The following inequalities can be proved by the analogous arguments as in the proof
of Theorem 4.1:
O, t 1;
z(m)(t)
z’(t) >0,
with
z (m-l)
>0, t >_t 2 >_tl,
z(t) defined by (4.7). We have z(t) > 0 for t >_ t I
and
z(t- r) < z(t) < y(t) + (t)z(t- r), t >_ t2,
[1- (t)]z(t- r) <_ y(t), t >_ t 2.
Choose a t* > t 2 such that
z(t-r) > O, t>_t*.
Since f(y) is increasing, we obtain
0
>_ z(m)(t) + Q(t)f(y(t o)
>_ z(m)(t) + Q(t)f[(1 i(t er)]z(t
r
er)), t _> t*.
Note that since z(t*-v- r) < z(t- t-r) for t > t*, we have
z(m)(t) + Q(t)f([1
where c
A(t r)]c) _< O,
t
>_ t*,
z(t*-r- r) > O. Integrating the above inequality from t*
to
t(t > t*),
we
get
z (ra
z (m
1)(t*) + / Q(s)f([1
A(s r)]c)ds <_ O.
t*
This leads to a contradiction in view of (4.10), since z (m> 0 for t >_ t 2. This
proves the assertion (i).
We can prove assertion (ii) and (iii) by the same arguments as in the proof of
Theorem 4.1. This completes the proof.
Theorem 4.3: Assume that f(- y) -f(y) .for y E
and that (4.4) and (4.5)
hold. If there exists a monotonically increasing function
+ oo), (0, + oo)]
such that
1)(t)
R+ cl[[t0,
+oo
[e(s)Q(s)(1 (s-r))-c’(s)]ds
+oo
(4.11)
> O, then conclusions (i)-(iii) of Theorem 4.1 remain true.
Proof: Let y(t) be an eventually positive solution of the inequality (4.1). Then,
there exists a t I >_ t o such that
for
any number c
y(t) > O, y(t- r) > O and y(t-er) > O for all t >_ t 1.
The following inequalities can be proved by the analogous arguments as in the proof
Oscillation Criteria
of Theorem 4.1:
__
tl;
A(t a)]z(t a) _< O,
t
for High Order Delay PDEs
z(’)(t) _< o,
Z (m-1)(t) > O,
z(t) > o,
Z’(t) > O,
z(m)(t) + eQ(t)[1
t
t2
201
>_ tz.
Thus, there exists T > t 2 such that z(T- r) > 0 and
Z
(m-
1)(t) _( Z (m- 1)(T),
z(m)(t) + ez(T
Set
t
>_ T;
(4.12)
r)Q(t)[1 (t r)] _< 0, t >_ t.
(4.13)
(m 1)
(t).
(t)_(t).z
z(T r)
then we obviously have
(t) > 0 for all t >_ T.
Note that
obtain
(t)
is a montonically increasing functions and using
9’(t)
<
z(m
z(T
(4.12)
and
(4.13),
we
’(t)z(m-1)(t) + (t)z(m)(t)
z(T r)
1)(
T) )’(t) + (t)
z(T (r)
ez(T r)Q(t)[1 A(t r)]
z(T )
t
> T.
Set
z(m-1)(T) =c>0;
z(T (r)
we have
’(t) <_ -[e(t)Q(t)(1 A(t r)) c’(t)], t >_ T.
Integrating both sides to the above inequality from T to
(t) <_ (T)
/ [e(s)Q(s)(1
t(t > T),
we get
A(s r)) c’(s)]ds,
T
which is impossible in view of assumption (4.11). This proves assertion (i).
We can prove assertion (ii) and (iii) by the same arguments as in the proof of
Theorem 4.1. The proof of Theorem 4.3 is complete.
Theorem 4.4:
Assume that A(t) A constant > O, f(- y)
f(y) > 0 for
and that f(y) is an increasing function and satisfies:
y
R+
f(x + y) <_ f(x) + f(y), f(kx) <_ kf(x) for x > O, y > O, k > O.
If Q(t)
is periodic with period
and
satisfies
(4.14)
XINZHI LIU and XILIN FU
202
Q(s)ds
+ oo,
(4.15)
then conclusions (i)-(iii) of Theorem 4.1 remain true.
Proof: Let y(t) be an eventually positive solution of the inequality
there exists a t I > t o such that
(4.1). Then,
y(t) > O, y(t- 7") > O and y(t-cr) > O for all t >_ t I
and for
v(t) + v(t- ).
z(t)
we have
z(m)(t) <_ O,t _> tl;
z’(t) >0, z (m-1)(t) > O, t_>t 2_>t 1.
z(t) > O,
Set
a(t)
Then, there exists
z(t) + z(t- 7")
a t3
y(t) + 2Ay(t- 7") + A2y(t 2v), t >_ t 2.
(4.16)
> t 1 such that
.(t) > 0..(t- ) > 0..’(t) > 0. t >_ t
and
ce
(m-1)(t)>0, c (m-1)(t-7")>0, t>_t3.
From (4.1) and (4.16)it follows that
.(")(t) v(")(t) + v()(t- ) + [v(’)(t- ) + v(")(t- 2)]
<_
Choose T
Q(t)f(y(t r)) Q(t r)f(y(t
7-
r)).
(4.17)
>_ t 3 such that
y(t- 27"- (r) > O, t >_ T.
(4.14), (4.16) and (4.17)"
a(m)(t) + a(m)(t r) + Q(t)f(a(t r))
< Q(t)f(y(t a)) 2AQ(t v)f(y(t v a)) A2Q(t 2v)f(y(t 2"
+ Q(t)f(y(t a) + 2Ay(t r a) + A2y(t 2r a))
< Q(t)f(y(t ag)) 2AQ(t)f(y(t r r)) A2Q(t)f(y(t 2r a))
+ Q(t)f(y(t )) 2AQ(t)f(y(t r r)) + A2Q(t)f(y(t 2r )) O, t >_ T.
(4.18)
Since a and f are increasing, we have
0 < c(T- (r) < a(s- a), s > T
Since Q(t)is periodic with period r, we get by
and
I(a(T- )) <_ f(a(s- a)),s >_ T.
Oscillation Criteria
Integrating both sides of
>_ ce (m
0
1)(t)
(4.18) from T to t(t > T),
f
c( m
-
,kce (m
1)(T
7")
7") c( TM
1)(T
7")
7")
Q(s)f(a(s-a))ds
T
1)(t
203
get
we
1)(T) -t- c (m 1)(t
ce (m
+
_> c (m
for High Order Delay PDEs
1)(T
ce(
TM
+ f(a(T- r))
1)(t
/ Q(s)ds.
T
c(m-1)(t)>0
and
This leads to a contradiction in view of (4.15), since
(m
Ce
7") > 0 for t _> t 3. This proves assertion (i).
We can proves assertion (ii) and (iii) by the same arguments as in the proof of
Theorem 4.1. This completes the proof of Theorem 4.4.
We shall consider next the case of A(t)< 0. The following lemma is a special
case of Theorem 2 in [3].
Lemma 4.2: [3] Assume that fl E C[[t 0, + o),R +] such that
--1)(t-
lim inf
_
/3(s)ds >-
(4.19)
t-5
and
limt__,+inf
/
t_
Then, the inequality
fl(s)ds > O.
x(m)(t)- m(t)x(t- mS) <_ 0
has no eventually negative bounded solutions.
We introduce the following notations:
cQ(t)
o
+ >
Theorem 4.5: Assume that the condition (4.5) holds, > 7", f(-y)- -f(y)
+, and that there exist constants ,kl,,k 2 and M such that
for y
1
<_ "1 /(t) /2 < O,
and
Q(t) >_ M > O, t >_ t o
if
t
>_ t o
(4.21)
XINZHI LIU and XILIN FU
204
limt__,+ooinf
J
fl(s)ds > 1,
t-5
then conclusions (i) and (iii) of Theorem 4.1 remain true.
Proof: Let y(t) be an eventually positive solution of the inequality
there exists a t I >_ t 0 such that
(4.1). Then,
y(t) > O, y(t- r) > 0 and y(t- r) > 0 for all t >_ t 1.
Set
y(t) + ;(t)y(t- r).
z(t)
We have
z(m)(t) <_
-Q(t)f(y(t- r)) <_ -cq(t)y(t- r) <_ O, t >_ t 1.
We claim that
z(t) < O,
If true, from
(4.1)
t
>_ t 1.
(4.23)
it follows that
z(m)(t) <_ eQ(t)y(t r) <_ eMy(t r), t <_ t 1.
(4.24)
Thus, we see that z (m- 1)(t) is strictly decreasing on (tl, + oo) and z(i)(t) are strictly
It, + c),
monotonically functions on
0, 1,..., m
lim z (mor
1)(t)
lim z (m
If
1)(t)
2. Then, we have
(4.25)
oo
r < + oe.
(4.26)
(4.25) holds, then we have
lim
z(i)(t)
c,
O, 1,..., m
1.
Hence (4.23) is true.
If (4.26) holds, then integrating both sides of (4.24) from t I to t and letting
t---, + cx, we get
eMy(s- a)ds <_ z( m
1)(tl)- r/,
(4.27)
I
which implies that y E
Ll[tl, + oo).
In view of (4.2),
zE
we obtain
Ll[tl, -t- oo).
Note that z(t) is montonically function,
lim
we see that
z(t)
O.
(4.28)
Oscillation Criteria
Thus r/- 0. From
(4.28),
for High Order Delay PDEs
it follows that
z(i)(t)z (i+l)(t)<O,
Equations
(4.28)
205
and (4.29)imply that
i-0,1,...,m-1, t>t 1.
(4.29)
(4.23)is true.
Now we have
v() <
which implies that
y(t)
()v(- ) <
v(- ) < v(- ),
is a bounded function. Thus
z(t)is bounded. Since
+) (
+ )(t ) + (t
>_ (t- + )(t- ) o t >_
z(t
+ -)
we h&ve
Q()
+ r) < Q(t)y(t- ),
(t-+r
From (4.24) and (4.30), it follows that
).z(t-a
z()(t) -(t- + )
)
t
> t 1.
(4.30)
z(t-(q-v)) < O, t>tl,
z(m)(t)- flm(t)z(t- mS)
O, t
t 1.
(4.31)
In view of (4.22), by Lemma 4.2 we see that the inequality (4.31) has no eventually
negative bounded solutions, which contradicts the fact that z(t)< 0 and z(t) is
bounded. This proves assertion (i). We can prove assertion (ii) and (iii) by the
same arguments as in the proof of Theorem 4.1. The proof is therefore complete.
5. Further Oscillation Criteria
In this section we shall establish some further oscillation criteria for the higher order
delay hyperbolic boundary value problem (2.1) under (B1)and (2.1) under (B2) using
the results obtained in the last two sections.
Theorem 5.1: Assume that conditions (U) and (4.5) hold, and that 0 <_ ;(t) <_ 1.
II
min_q(x,s)[1-)(s-)]ds-
x
then
a omio. of
.
(5.1)
(B) a ocilmou i. a.d
ro6tm (2.)
the
under
(B2) are oscillatory in G.
of problem (2.1)
Proof: Let u(x,t) be a nonoscillatory solution of the problem (2.1) under (B1).
We may assume that u(x,t)>O for (x,t) Efx[#,+c), where # is a positive
number. By the analogous arguments as in the proof of Theorem 3.1, we can see
that the function U(t) defined by (3.6) is a positive solution of the inequality (3.1) for
t >_ t o >_ #, which implies that the function U(t) defined by (3.6) also is a positive
(i)
(ii)
all solutions
XINZHI LIU and XILIN FU
206
solution of the inequality
d+ A(t)U(t
m[U(t)
v)] + min_ q(x, t)f(U(t )) _< 0.
(5.2)
zfl
However, by Theorem 4.1, we see that the inequality (5.2) has no eventually positive
solutions. Thus, we obtain a contradiction.
If u(x,t)<O for (x,t)x[#,+c), then
-u is an eventually positive
solution of the problem (2.1) under (B1) which is impossible. This proves assertion
(i).
The assertion (ii) can be proved by the analogous arguments as in the proof of
(i). The proof of Theorem 5.1 is complete.
Using Theorem 4.2-4.5, respectively, it is easy to obtain the corresponding results
for problem (2.1) under (B)or (2.1) under (B2)also. We merely state them below.
Theorem 5.2: Assume that the condition (H) holds, and that 0 < (t) < 1, f(y)
is a monotone increasing function in R +. If for any c > O,
assertion
min_ q(x, s)f([1
A(s r)]c)ds
+ oc,
then
all solutions
all solutions
of the problem (2.1)
of the problem (2.1)
under
under
(B1)
(B2)
are oscillatory in
are oscillatory in
G and
G.
Theorem 5.3: Assume that conditions (H) and (4.5) hold, and that 0 <_ A(t) <_ 1.
exists a monotonically increasing function E CI[R
there
If
+ (- ,-t-cx)] such that
[(s)min_ a(x, s)(1 i(s )) c’(s)]ds
4- c,
xE
(5.4)
any number c > O, then
all solutions of the problem (2.1) under (B1) are oscillatory in G and
(i)
(ii) all solutions of the problem (2.1) under (B2) are oscillatory in G.
Theorem 5.4: Assume that condition (H) holds, A(t) =_
constant > O, and
that f(y) is an increasing function and satisfies (4.14). /f q(x,t) is periodic in t with
period r and satisfies
for
.
+oo
min_q(x,s)ds-+oc,
(5.5)
x
then
all solutions of the problem (2.1) under (B1) are oscillatory in G and
all solutions of the problem (2.1) under (B2) are oscillatory in G.
Theorem 5.5: Assume that conditions (H) and (4.5) hold, > v, and that there
exist constants "1,’2, and M such that
(i)
(ii)
-l
<_
and
min
xEft
_< (t) _< 2 <0,
tER+
q(x,t) > M > O, t R +.
lirinf+
fl (s)ds >
1,
Oscillation Criteria
for High Order Delay PDEs
207
then
()
all solutions
all solutions
(ii)
of the problem (2.1) under (B1)
of the problem (2.1) under (B2)
are oscillatory in
are oscillatory in
G and
G,
where
cmin_ q(x, t)
xEft
’(t)
-a(t-+-)
and
m
Finally, we discuss two examples.
Example 5.1" Consider the equation
Ot6 + (1 -e )u(x,t- -)] + 3u + 2u(x,t- )exp[3t + x +
u2(x,t-)]
Xu + (2 +
),
os t)x(, t
(, t) e (0,,) (0, + )
(B1)
,,(o, t) + (o, t) 0, (,, t) + ,,(,, t)
and a boundary condition of type
Here,
A(t)
m- 6;n-
1
e
t;
o, t > o.
;7(x,t)- 1
X;,g- 1;gt- (0,’);r- ’;a
(5.8)
for
q(t)--2eat; f(u)--ue u
q(z,t)--2e z +3t,min
:
t>0;
2
[o,]
It is easy to see that the function f(u) satisfies condition
(H)
-boo
/
Then, all conditions of Theorem 5.1
are fulfilled.
(5.7)
and (5.8) are oscillatory in(0, rr) x (0,
Example 5.2: Consider the equation
(9 4
-[u
and a boundary condition of the type
,,(o, t)
f(u)
m
u.
4;n
1;e
and
4(t)-
+ oo.
Hence, all solutions of problems
sin x)u(x, t
4’)
(x, t)i(O, 7r) (0, q- oo)
(B2)
1;f2 -(0, 7r); A(t)
5
gds
(5.9)
),
u(,, t)
In this case,
_s+ n"
+ oo).
u(x, t 2r)] + 2u + 4(2
etAu -k 3An(x, t
Here,
2e 3s. e
m.in .q(x,s)[1-A(s-r)]ds-
and
r- r
m
o, t > 0.
1;r
2rr;r
rr
--2
emin q(x, t)
e [0,,]
A(t- O" q- 7")
71"4’
(5.10)
4rr;q(z,t)
r4(2 sinx);
XINZHI LIU and XILIN FU
208
where e
1. It is easy to see that
lim
fl (s)ds
inf
t-
+ oo
t-r
rds
lim inf
t
+ oo
_"
2
-
> -.
The hypotheses of Theorem 5.5 are satisfied and hence all solutions of the problem
(5.9) and (5.10) are oscillatory in (0, r) x (0, / c).
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--i-+dmu