DELAY ORDER

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 ofhighorder delay partial differential equations.

Employing high order delay differential inequalities, several oscillation cri- teria are established for such equations subject to two different boundary conditions.

Two

examples arealso given.

Key

words:

Oscillation,

Higher Order Delay Partial Differential

Equa-

tions, Differential Inequality, Eventual Positive Solutions.

AMS

subjectclassifications: 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 ex- cellent exposition of the basic theory, see

[5]. ^In

recent years, there has been an in- creasing interest in oscillation theory ofdelay partial differential equations, ^see

[6-10]

and references therein.

However,

the corresponding theory is still in its initial

stage

of development.

In

this paper, weshall investigate a class ofhigh order delay partial differential equations which will bedescribed in Section 2.

In

Section

3,

weshall esta- blish several oscillation criteria for high order delay partialdifferential equations sub- ject to two kinds ofboundary

conditions,

employing

Green’s

theorem and high order delay differential inequalities.

We

then develop, in Section

4,

some results on even- tual 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

illu- strateour

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.()1998by North Atlantic SciencePublishingCompany 193

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

^r))

= a(t)Au + E aj(t)Au(x, aj(t)),(x,t)e x 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;

A

E

cm[R ₊ ,R];

_{a, aj}

C[R ₊ ,R + ^],

^j

^{1, 2,...,} ;

^p,q

C[R ₊

^x

,R + ^], f C[R, R],

_rj

C[R

_+,

R +

^is nondecreasing in

t, rj(t) ^<

^{t and lim}

^r(t) +

^cx3,

j

1, 2,..., .

^t--.

⁺

We

shall consider two kinds ofboundaryconditions

Ou )u o, (, ) e ^{O R} ₊

and

ON ^t- 7(x’ (B1)

0, (, ) e 0e a

_+,

_(B)

where

N

is the unit exterior normal vector to

0, 7(x, t)

^{is a}nonnegative continuous functionon 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 apoint

(Xo, to)

^{2 x}

^lit, + cx)

^{such that}

U(Xo, to) ^O.

3. Oscillation Criteria

In

this section we shall establish oscillation criteria for problem

(2.1)

^{with boundary}

condition

(B1)

^and

(B2)

separately. The basic idea ofour 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,

^uE

R + If

^{the high order delay}

differential

inequalities

d--t-[U(t

d,

^{+ (t)U(t r)] + P(t)U(t) + Q(t)f(U(t o’)) _<}

⁰

^(3.1)

has no eventually positive

solutions,

^then all solutions

of

^{the problem}

(2.1)

^under

(B1)

are oscillatory in

G,

where

P(t)

min_

p(x, t), Q(t)

min_

q(x, t).

xEfl xEfl

Proof:

Let u(x,t)

^{be a} nonoscillatory solution of the problem

(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 to

>

it, such that

and

(,

^t

-) > 0, (,

^t

) >

⁰

u(x,rj(t)) ^{> O, (x,t) e ]x[t}

^o,

^+oo),

^{j- 1,2,...,g.}

Oscillation Criteria

for

^{High Order Delay}

^PDEs

¹⁹⁵

Integrating both sides ofsystem

(2.1)

^{with respectto x over thedomain}

f2,

weobtain

dm u(x, t)dx + ^a(t) u(x

^t

r)dx + ^{p(x, t)u(x t)dx}

dt^m

+ J ^{q(x, t)f(u(x,}

^t

^r))dx

a(t)/Au(x, ^{t)dx + E aj(t)J Au(x,}

^a

^j(t))dx,

^t

_

^t0.

t ^j=l Ft

From Green’s Theorem,

^itfollows that

and

/ /

-/7(,j(t))(,y(t))d8 ^{<_ o,}

^j-

^1,2,...,t,

^t

^{>_ to,}

o

(3.3)

(3.4)

where dS is the surface integral element on

0ft.

using

Jensen’s

inequality, we have

Since

if(u)

^{is convex in}

^R

+, then

f (u(x,

^t

^)dx _ _{all [o1} ^u(x,

^t

^{)dx (3.)}

where

a] f

^{dx. Combining}

(3.2)-(3.5)

yields

+ P(t) u(x,t)dx + ^Q(t)f _a

_ ^a(t) _/ 7(x, t)u(x, t)dS E.aj(t) / ^7(x, j(t))u(x, aj(t))dS

<_O, t>_t

_o.

Thus,

^{we see} that thefunction

1

/ ^{u(x t)dx}

u(t) al ^(3.6)

is a positive solution ofthe inequality

(3.1)

^{for t}

_> to,

which contradicts the condition of the theorem.

If

u(x, t) <

0 for

(x, t) [#, + c),

^{then set}

(., t) (., t), (., t) [, + ).

Note

that since

f(-u)= -f(u),

^uE

(0, + ^c),

^{it is easy to check that}

^(x,t)is

^a

positive solution of the problem

(2.1)

^under

(B1)

which is impossible. This com- pletes the proofofTheorem 3.1.

The following fact will be used in the proof of Theorem3.2. Consider the Dirich- let problem

Au+u-0 in

,

ulo-O,

where

,-

constant.

It

is well known that the smallest eigenvalue

0

^{and the cor-}

responding eigenfunction

(x)

^{are positive.}

Theorem 3.2:

Assume

that the condition

(H)

^holds.

If

the high order delay

differential

^inequality

dm [V(t) + A(t)V(t v)] + (A0a(t) + P(t))V(t) + ^Q(t)f(V(t r)) <

⁰

(3.7)

dt^m

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 to

>

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

(I)(x)

^and integrating withrespect to x over the domain

fl,

^{we have}

d^rn

+ I ^{p(x, t)u(x, t)qP(x)dx + / q(x, t)f(u(x,}

^t

^(r))O(x)dx

a(t)J Au(x, t)(x)dx + E aj(t)] Au(x,rj(t))(x)dx,

^t

^>

^tO.

(3.8)

fl 3--1

Using

Green’s Theorem,

^{we obtain}

Au(x, t) ((x)dx

Oscillation Criteria

for

^{High OrderDelay}

^PDEs

¹⁹⁷

= / ^(3.9)

Au(x, crj(t)). ^(b(x)dx

o i ^u(x, rj(t))(b(x)dx,

^j

^{1, 2,...,} . ^(3.10)

Using

Jensen’s

inequality, ^wehave

f(u(x,

^t

cr))O(x)dx

>_ (x)dx. f

f ^((x)dx

Combining

(3.8)-(3.10)

^yields

(3.11)

- ^{Ja(t) i} u(x, t)((x)dx,

^t

>_ to,

i.e.,

the inequality

(3.7)

^has positive solution

f

] u(x, t)((x)dx,

^t

>_ to, v(t) r

_J

_((x)dx

which contradictsthe condition of the theorem.

If

u(x, t) <

^{0 for}

(x, t)

^{E x}

[#, + oc),

^{then -u is a} positive solution of the problem

(2.1)

^under

(B2)

^{which also provides a} contradiction. The proof of Theorem 3.2 iscomplete.

4. High Order Delay Differential Inequalities

From

the discussion in Section 3 it followsthat the problem ofestablishingoscillation criteria for the system

(2.1)

^{can be reduced to the} investigation ^of the properties of the solution ofhigh order delay differential inequalitiesfor the form

-m[y(t)

m

^{+ (t)y(t 7")] + Q(t)f(y(t or)) <_ O,}

^t

^{>_ to, (4.1)}

and d^m

d--[y(t ^{+ (t)y(t 7")] + Q(t)f(y(t r)) O,}

^t

^>_

^to.

(4.2)

Along

^with

(4.1)

^and

(4.2),

^{we consider} the high orderdelay differentialequation d^TM

dtm[y(t ^{+ (t)y(t ’)] + Q(t)f(y(t )) o,}

^t

^{>_ to, (4.3)}

where m is an even positive integer,

" ^>

^{0 and r}

^>

^{0 are}

^constants;

cm[[to,+ c), R], ^Q C[[,

0,

+ ^{), R} ₊

^{fo om}

*o ^{> 0, f e C[R, R]. We}

^h

first consider thecase

A(t) >_

^0.

Assume

that

y(t)

^isa nonoscillatory solution of equation

(4.3). ^Let z(t) y(t) + ^A(t)y(t-

We

shall usethe 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 >_

_to ^and an integer

k,

O

<_

k

<_

m, with rn

+

^{k even}

for

z(t)z(m)(t) ^{>_ O,}

^orrn

+

^{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.

Theorem4.1-

Assume

that

f(- ^y) f(y) for

^y

^R

+, and that

0

_< A(t) <_ 1, Q(t) >_ O,

t

>_ to; (4.4) f(Y)

_y

>

^e constant

> O,

y

(0, + oz) (4.5) If

Q(s)[1 A(s cr)]ds +

^cx,

(4.6)

then

(i)

the inequality

(4.1)

^{has no eventually}positive

solutions;

(ii)

^{the inequality}

(4.2)

^{has no eventually} negative solutions; and

(iii)

all solutions

of

^{the equation}

(4.3)

^are oscillatory.

Proof:

Let y(t)

be an eventually positive solution of the inequality

(4.1). Then,

there exists a

t ^{> to,}

^{such that}

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, (4.7)

wehave

z(t) > O,

t

>_

t_1.

From (4.1), (4.4)

^and

(4.5)it

follows that

Oscillation Criteria

for

^{High Order}

^{Delay PDEs}

¹⁹⁹

z(rn)(t) ^<_ -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 an odd number k and a t₂

_

^t1

such that

z()(t) ^{> o, o <_ <_} ,

^t

^>_ _t

and

(-1) i-kz(i)(t)>0,

^k

^{< <_}

^{m, t}

^>_

^t2.

It

is easyto see that

z’(t) > O,

z

(m-1)(t) > 0,

^t

_

^t2.

^(4.8)

Using

(4.5)and (4.7),

^{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.

Note z(t) >_ y(t)

for t

>_ _t2,

thus weobtain

0

>_ z(m)(t) + ^{eQ(t)[z(t r) A(t r)z(t}

^r

^r)],

^t

^>_

^t2.

Since

z(t)

^is increasing for ^t

>_ _t2,

^we have

z(m)(t) + ^eQ(t)[1 (t a)]z(t a) _< O,

^t

>_

t_2.

(4.9)

Integrating both sides of

(4.9)

^{from t}2 to

t(t > t2)

^we

^get

z(m

1)(t) _

^z(m

^{1)(t2) ez(t}

²

^r) _J ^{Q(s)[1 A(s r)]ds.}

2

Since z(m-

a)(t) >

^{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 positivesolution of

(4.1).

^{Theproofoftheasser-}

tion

(iii)

^{is obvious.}

Theorem4.2:

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,

(4.10)

then conclusions

(i)-(iii) of

Theorem 4.1 remain true.

Proof:

Let y(t)

be an eventually positive solution of inequality

(4.1).

there exists a t_I

>

^t_{o such}^that

Then,

y(t) > 0, y(t-r) >

^{0 and}

y(t-r) >

0 for all t

_> ta.

The following inequalities ^can be proved by the

analogous arguments

^{as in the} proof of Theorem 4.1:

z(m)(t) ^O,

^t

1;

z’(t) ^>0,

^z(m-l)

>0,

t

>_t

₂

>_tl,

with

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* >

t₂ such that

z(t-r) > O, t>_t*.

Since

f(y)

^isincreasing, 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*,

^wehave

z(m)(t) + ^Q(t)f([1 A(t r)]c) _< ^O,

^t

>_ t*,

where c

z(t*-r- r) > ^O.

Integrating the above inequality from

t*

to

t(t > t*),

^we

get

z(m _z(ra

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-}

1)(t) >

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

^yE

R+

^{and that}

^(4.4)

^and

^(4.5)

hold.

If

^{there exists a} monotonically increasing

function cl[[t0, ₊ oo), (0, + ^oo)]

such that

+oo

[e(s)Q(s)(1 (s-r))-c’(s)]ds +oo (4.11)

for

^{any numberc}

> 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

>_

to such that

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

for

^High Order Delay

PDEs

201 ofTheorem 4.1:

z(t) > o, z(’)(t) ^{_< o,}

Z’(t) > O,

^Z

(m-1)(t) ^{> O,}

^t

_

^t2

_ ^tl;

z(m)(t) + ^eQ(t)[1 A(t a)]z(t a) _< ^O,

^t

>_ tz.

Thus,

there exists

T >

t₂ such that

z(T- r) >

^{0 and}

Z(m-

1)(t)

_(^Z(m-

1)(T),

^t

>_ T; (4.12)

z(m)(t) + ^{ez(T r)Q(t)[1 (t r)] _< 0,}

^t

^>_

^t.

^(4.13)

Set

_{(m 1)}

(t)_(t).z ^(t).

z(T r)

thenweobviously have

(t) >

^{0 forall t}

>_ T.

Note

that

(t)

^{is a} montonically increasing functions and using

(4.12)

^and

(4.13),

^we

obtain

’(t)z(m-1)(t) (t)z(m)(t)

9’(t)

z(T r) + _z(T

(r)

z(m 1)( ez(T r)Q(t)[1 A(t r)]

< z(T T) ^{)’(t) + (t)} z(T )

^t

> T.

Set

we have

z(m-1)(T)

z(T (r) ^=c>0;

’(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 > T),

^we

^get (t) <_ (T) / [e(s)Q(s)(1 A(s r)) c’(s)]ds,

T

which is impossiblein view of assumption

(4.11).

This proves assertion

(i).

We

can prove assertion

(ii)

^and

(iii)

^{by the same}

^arguments

^{as in the proofof}

Theorem 4.1. The proofofTheorem 4.3 iscomplete.

Theorem 4.4:

Assume

that

A(t) ^A

^constant

> O, f(- y) f(y) >

⁰

for

y

R+

^andthat

^f(y)

^{is an increasing}

^function

^and

^satisfies:

f(x + ^{y) <_} f(x) + ^f(y), f(kx) <_ kf(x) for

^x

> O,

y

> O,

k

> ^O. (4.14)

If ^Q(t)

^isperiodic with period and

satisfies

Q(s)ds +

^oo,

(4.15)

then conclusions

(i)-(iii) of

^{Theorem4.1 remain true.}

Proof:

Let y(t)

be an eventually positive solution of the inequality

(4.1). ^Then,

there exists a t_I

>

to ^{such that}

y(t) > O, y(t- 7") > O

and

y(t-cr) > ^O

^{for all t}

>_

^t_I

and for

z(t) v(t) + v(t- ).

we have

z(t) > O, z(m)(t) ^<_ O,t _> _tl;

z’(t) >0,

z

(m-1)(t) > O, t_>t2_>t

1.

Set

a(t) z(t) + z(t- 7") y(t) + ^2Ay(t- 7") + A2y(t 2v),

^t

>_

^t_2.

(4.16)

Then,

there exists at₃

>

^t₁^{such that}

and

.(t) > 0..(t- ) > 0..’(t) > 0.

^t

^>_ t

ce

(m-1)(t)>0,

^c

(m-1)(t-7")>0, t>_t

_3.

From (4.1)

^and

(4.16)it

follows that

.(")(t) v(")(t) + v()(t- ) + [v(’)(t- ) + v(")(t- 2)]

<_ Q(t)f(y(t r)) Q(t r)f(y(t

^7-

r)). (4.17)

Choose

T >_

^t₃ such that

y(t-

27"-

(r) > O,

t

>_ T.

Since

Q(t)is

^{periodic with} period r, we

get

by

(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, wehave

0

< c(T- (r) < a(s- a),

^s

> T

and

I(a(T- )) <_ f(a(s- a)),s >_

^T.

Oscillation Criteria

for

^{High Order Delay}

^PDEs

²⁰³

Integrating both sides of

(4.18)

^from

^T

^to

t(t > T),

^we

^get

0

>_

ce(m

1)(t)

^ce(m

1)(T) -t- c

^(m

1)(t 7")

^,kce(m

1)(T 7")

+ f Q(s)f(a(s-a))ds

T

_>

^c(m

1)(t ^c(

^m

⁺ 1)(T ^f(a(T- -- ^{ce( r))}

^TM

^/

T

^{1)(t Q(s)ds. 7") c(}

^TM

^{1)(T 7")}

This leads to a contradiction in view of

(4.15),

^since

c(m-1)(t)>0

^and

Ce(m

--1)(t- 7") >

^{0 for t}

_>

^t_3. This proves assertion

(i).

We

can proves assertion

(ii)

^and

(iii)

^{by the same}

^arguments

^{as in the proofof}

Theorem 4.1. This completes the proof of Theorem4.4.

We

shall consider next the case of

A(t)<

^{0. The} following lemma is a special case ofTheorem 2 in

[3].

Lemma

4.2:

[3] ^Assume

^that

fl

^E

C[[t

0,

+ o),R +]

^{such that}

and

Then,

^the inequality

lim inf

/3(s)ds >-

t-5

limt__,+inf / ^{fl(s)ds > O.}

t_

_

x(m)(t)- m(t)x(t- mS) <_

⁰

(4.19)

has no eventually negative bounded solutions.

We

introducethe followingnotations:

cQ(t)

+ ^> o

Theorem 4.5:

Assume

that the condition

(4.5) holds, >

7",

f(-y)- -f(y) for

^y +, and that there exist constants

,kl,,k

2 and

M

such that

1

<_ "1 ^/(t) /2 ^{< O,}

^t

^>_

^to

(4.21)

and

Q(t) >_ M > O,

t

>_

to

if

limt__,+ooinf J

_t-5

^{fl(s)ds > 1,}

then conclusions

(i)

^and

(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

>_

t₀ such that

y(t) > O, y(t- r) >

^{0 and}

y(t- r) >

0 for all t

>_

t_1.

Set We

have

z(t) y(t) + ^;(t)y(t- r).

z(m)(t) ^<_ -Q(t)f(y(t- r)) <_ -cq(t)y(t- r) <_ O,

t

>_

t_1.

We

claim that

z(t) < O,

^t

>_

t_1.

(4.23)

If

true,

from

(4.1)

^{it follows that}

z(m)(t) ^{<_ eQ(t)y(t r) <_ eMy(t r),}

^t

^<_

^t1.

(4.24)

Thus,

we see that z(m-

1)(t)

^is strictly decreasing on

(tl, + ^oo)

^and

z(i)(t)

^{are strictly}

monotonically functions on

It, + c), 0, 1,...,

^{m 2.}

Then,

^we have

lim z(m-

1)(t)

^oo

(4.25)

or

lim z(m

1)(t) r ^< +

^oe.

(4.26)

If

(4.25) holds,

then wehave

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

^viewof

(4.2),

^{we obtain}

z E

Ll[tl, -t- oo).

Note

that

z(t)

^ismontonically

function,

we see that

lim

z(t) ^O. (4.28)

Oscillation Criteria

for

^{High Order Delay}

^PDEs

²⁰⁵

Thus _r/- 0.

From (4.28),

^it follows that

z(i)(t)z(i+l)(t)<O,

i-0,1,...,m-1, t>t_1.

Equations

(4.28)

^and

(4.29)imply

that

(4.23)is

^true.

Now

wehave

(4.29)

v() < ()v(- ) < v(- ) < v(- ),

which implies that

y(t)

is a boundedfunction. Thus

z(t)is

^{bounded. Since}

z(t + ) ( + )(t ) + (t + -)

>_ (t- + )(t- ) o

t

>_

weh&ve

Q() ).z(t-a ^{+ r) < Q(t)y(t- ),}

^t

^>

^t1.

^(4.30)

(t-+r

From (4.24)

^and

(4.30),

^itfollows that

) ^{z(t-(q-v)) <} O, t>tl,

z()(t) -(t- + )

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 inthe proofofTheorem4.1. The proofis thereforecomplete.

5. Further Oscillation Criteria

In

this section we shall establish some further oscillation criteria for the higher order delay hyperbolic boundary valueproblem

(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- (5.1)

x

then

(i) a

omio.

of ro6tm (2.) . ^(B)

^a

^ocilmou

^{i. a.d}

(ii)

all solutions

of

^{the problem}

(2.1)

^under

(B2)

^are oscillatory in

G.

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

>_

_to

>_

#, which implies that the function

U(t)

^{defined by}

(3.6)

^{also is a positive}

solution of the inequality

m[U(t)

d-

^{+ A(t)U(t v)] +}

_zfl^min_

^{q(x, t)f(U(t )) _<}

^0.

^(5.2)

However,

by Theorem

4.1,

we see that the inequality

(5.2)

^{has no eventually positive}

solutions.

Thus,

we obtain acontradiction.

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 assertion

(i).

^{The proofof}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 statethem 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,}

min_

q(x, s)f([1 A(s r)]c)ds +

^oc,

then

all solutions

of

^{the problem}

(2.1)

^under

(B1)

^are oscillatory in

G

and allsolutions

of

^{the problem}

(2.1)

^under

(B2)

^are oscillatory in

G.

Theorem 5.3:

Assume

that conditions

(H)

^and

(4.5) ^hold,

^{and that 0}

<_ A(t) <_

^1.

If

^{there exists a} monotonically increasing

function

^E

CI[R + ^(- ,-t-cx)]

^{such that}

[(s)min_ a(x, s)(1 i(s )) c’(s)]ds

4-c,

(5.4)

xE

for

^{any number c}

> O,

^then

(i)

all solutions

of

the problem

(2.1)

^under

(B1)

^are oscillatory in

G

and

(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)

isperiodic in t with period^r and

satisfies

+oo

min_q(x,s)ds-+oc, (5.5)

then ^x

(i)

all solutions

of

^{the problem}

(2.1)

^under

(B1)

^are oscillatory in

G

and

(ii)

^allsolutions

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}

and

-l

<_ _< (t) _< 2 ^<0, tER+

min

q(x,t) > M > O,

^t

R _+.

xEft

inf

fl (s)ds > 1,

lir +

Oscillation Criteria

for

High Order Delay

PDEs

207 then

() (ii)

where

and

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,

cmin_

q(x, t)

’(t) -a(t-+-)

xEft

m Finally, wediscuss two examples.

Example 5.1" Consider theequation

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, + )

and a boundarycondition of type

(B1)

,,(o, t) + (o, t) 0, (,, t) + ,,(,, t) o,

t

> o.

Here,

^m-6;n-

X;,g-

1;gt-

(0,’);r-

’;a

;7(x,t)-

^{1 for}

A(t)

^{1 e}

t;

q(z,t)--2e

^z

+3t,min q(t)--2eat; f(u)--ue

^u2 : [o,]

It is easy to see that the function

f(u)

satisfies condition

(H)

^and

-boo

/ m.in .q(x,s)[1-A(s-r)]ds- Then,

all conditions of Theorem 5.1 are fulfilled.

(5.7)

^and

(5.8)

^are oscillatory

in(0, rr)

^x

(0, + ^oo).

Example 5.2: Consider the equation (9⁴

-[u ^u(x,

^t

^{2r)] +}

^2u

^{+ 4(2}

^sin

^x)u(x,

^t

^4’)

etAu

-k

3An(x,

^t

), ^{(x, t)i(O, 7r) (0,}

^q-

^oo)

(5.8)

t>0;

_s+^n"

2e3s._e

gds +

^oo.

Hence,

all solutions of

problems

(5.9)

and aboundary condition ofthe type

(B2)

,,(o, t) u(,, t) o,

t

>

0.

(5.10)

Here,

^m

4;n 1;e 1;f2 -(0, 7r); A(t) 1;r 2rr;r 4rr;q(z,t) r4(2 sinx);

f(u)

^u.

In

thiscase,

5 ^r-_m^{r rr} and --2

emin

q(x, t)

e[0,,]

4(t)-

A(t-

^O"q-

7") 71"4’

where e 1.

It

is easy to see that

inf

fl (s)ds

^{lim inf rds}

- ^{> -.}

limt-

+

^{oo t}

+

^oo

t-r _"

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).

References

[4]

[7]

[8]

[9]

[10]

Fu, X.

and

Liu, X.,

Oscillation criteriafor aclass ofnonlinear neutral parabolic partial differential equations, Appl. Anal. 58

(1995),

^215-228.

Georgou D.

and Kreith,

K.,

Functional characteristic initial value problems,

J.

Math. Anal. Appl. 107

(1985),

^414-424.

Grace, S.R.

and Lalli,

B.S.,

Oscillation theorems for certain delay differential inequalities,

J.

Math. Anal. Appl. 106

(1985),

^414-426.

Kiguradze,

I.T., On

the oscillation of solutions of the equation

--i-+dmu

a(t) lu ^{nsignu = 0, Mat.}

^{Sb. 65}

^(1964),

^172-187.

(in Russian)

Ladde, G., Lakshmikantham, V.

and

Zhang, B.G.,

Theory

of

Oscillation

for Differential

Equations with Deviating

Arguments,

Marcel

Dekker, Inc., New

York 1987.

Liu,

X.

and

Fu, X.,

^Nonlinear differential inequalities withdistributed deviating

arguments

and applications, Nonlinear World1

(1994),

^409-427.

Liu,

X.

and

Fu, X.,

Oscillation criteria for nonlinear inhomogeneous hyperbolic equationswith distributed deviating

arguments, JAMSA

^9:1

(1996),

^21-32.

Mishev,

D.P.

and Bainov,

D.D.,

Oscillation propertiesof the solutions of a class of hyperbolic equations of neutral type, Funkcial. Ekvac. 29

(1986),

^213-218.

Mishev, D.P.

and

Bainov, D.D.,

Oscillation of the solutions of parabolic differential equations of neutral type, Appl. Math.

Comput.

28

(1988),

^97-111.

Yoshida, N.,

Oscillation of nonlinear parabolic equations with functional

arguments,

^Hiroshima Math.

J.

16

(1986),

^305-314.