Forced oscillations of solutions of impulsive nonlinear parabolic differential-difference equations

J. Korean Math. Soc. 35 (1998), No. 4, pp. 881-890

FORCED OSCILLATIONS OF SOLUTIONS OF IMPULSIVE NONLINEAR PARABOLIC DIFFEHENTIAL-DIFFERENCE EQUATIONS

DRUMI BAINOV AND EMIL MINCHEV

ABSTRACT. Sufficient conditions for forced oscillations of the so- lutions of impulsive nonlinear parabolic differential-difference equa- tions are obtained.

1. Introduction

The impulsive differential equations are adequate apparatus for math- ematical simulation in the science and technology. These equations pro- vide natural mathematical description of processes which are subject to short-time perturbations during their evolution. In contrast to the big number of results for impulsive ordinary differential equations collected in seven monographs during the last "eight years [2J, [7J-[l1J, [15J, the first results for impulsive partial differential equations were obtained in the recent years [1], [3J-[6J, [12J-[14J.

The present paper is concerned with the forced oscillations of solu- tions of impulsive nonlinear parabolic differential-difference equations subject to certain boundary conditions. The oscillation properties of the solutions are investigated via averaging technique.

2. Preliminary notes

Let 0 c jRn be a bounded domain with a smooth boundary 80. and

n⁼ 0 U 80. Suppose that 0"= to < tt < t2 < ... < tk < .. , are given

Received June 26, 1997.

1991 Mathematics Subject Classification: 35B05.

Key words and phrases: impulsive parabolic differential-difference equations, forced oscillation of solutions.

numbers andtk+l =tk+er, k= 0, 1, . " , whereer =const >

°

fixed natural number.

Define Jimp = {tk}k~=l' lR+ = [0,+(0), EO = [-er,O] x n, E =

(0, +(0) x n, E* ⁼ JR+ x n, Eimp ⁼ Ht, x) E E: t E Jimp}, Eimp = Ht, x) E E*: t E Jimp}.

LetCimp[EO^UE* ,JR] be theclassof all functionsu: EOUE* ^{- T}JRsuch that:

(i) The restriction of u to the set EO U E* \ Eimp is a continuous function.

(ii) For each (t,x) E Eimpthere exist the limits

lim u(q,s) =u(t-, x), lim u(q,s)=u(t+,x)

(q,s)-(t,x) (q,s)-(t,z)

q<t q>t

and u(t, x) =u(t+, x) for (t, x) E Eimp.

The class of functions Cimp[E*, JR] is defined analogouslyasE*iswrit- ten instead ofEOUE* in the above definition.

Consider the nonlinear parabolic differential-difference equation (1) Ut(t, x) - a(t)Llu(t, x)+p(t, x)j(u(t - er, x))

= H(t, x), (t,x) E E\ Eimp, subject to the impulsive condition

(2) u(t,x) - u(r,x) =g(t,x,u(r,x)), (t,x) E Eimp, and the boundary conditions

(3) or, (4)

auOn(t, x)+'Y(t, x)u(t, x)=0, (t, x) E (JR+ \ Jimp)^X an,

u(t, x) = 0, (t,x) E (JR+ \ Jimp )^X 00.

The functions a: JR+ -+ JR, p: E* ^-+ JR, j: JR -+ JR, H: E* ^{- T} JR,

g: Eimpx lR ^{- T}JR,'Y: JR+ x an ^{-+ JR}are given.

DEFINITION 1. The function u: EOUE* ^-+ JR is called a solution of the problem (1)-(3) ((1), (2), (4)) if:

(i)uE Cimp[EOuE*,JR],there exist the derivatives ut(t,x), ux;x;(t, x), i =1, ... ,n for (t,x) E E \ Eimp^and u satisfies (1) on E \ Eimp.

(ii) u satisfies (2), (3) ((2), (4)).

Forced oscillations of solutions 883

x <0.

x >0, x=O,

DEFINITION 2. The nonzero solution.u(t,x) of equation (1) is said to be nonoscillatingifthere exists a number f..L ~

°

constant sign for (t,x) E [f..L,+(0)x O. Otherwise, the solution issaid to oscillate.

FOTthe fun~;tionsign ^'1Nl:?: hageadopted the following aennition

signx= {

~:

-1 if Introduce the following assumptions:

HI. a E Cimp[lR+,lR+J.

H2. pECimp[E*,lR+J.

H3. 9E C(E:mp ^{x lR, lR).}

H4. 'Y E Cimp[lR+ ^X ao,lR+].

H5. f E C(lR,lR), f(u) = -f(-u) foru~ 0, f is a positive and convex function inthe interval (0,+(0).

H6. HE Cimp[E*,lRJ.

In the sequel the following notations will be used:

P(t) = min{p(t,x): x E O},

V(t) = [ u(t,x).b, ( [ dx)^{- I ,}

Ho(t) = [ H(t,x)dx ( [ dx) ^-I

3. Main results

We give sufficient conditions for oscillation of the solutions of problem (1 )-(3).

LEMMA 1. Let the following conditions hold:

1. AssumptionsH1-H6arefulfilled.

2. u E C2(E\Eimp)nCl(E*\E:mp)is a positive solution of the problem (1)-(3) in the domain E.

3.g(tk'x,~) ::;_Lk~, k= 1,2, ... ,xE 0,~ E lR+,Lk~

°

Then the function Vet) satisfies for t ~ ^(j the impulsive differential inequality

(5) (6)

V/et) +P(t)f(V(t - er)) ::; Ho(t), t i= _tk,

V(tk) ::; (1+Lk)V(tk).

(7)

Proof. Lett _~ er. Integrating the equation (1) with respect to x over the domainn, we obtain

~

1

1

n n

+

1

1

n n

From the Green formula and H4 it follows that

(8)

J

1

J

n an an

Moreover, for t i=tk, the Jensen inequality enables us to get

J

J

n n

(9) ? P(t)/ ( [ u(t - a,x)dx ( [ dX) - ') [ dx

~

In virtue of (8) and (9) we obtain from (7) that V/et) +P(t)f(V(t - er)) ::; Ho(t), Fort = tk we have that

V(tk) - V(tk) ::; ^Lk

(1

^f

~

n n '

that is,

o

Forced oscillations of solutions 885 DEFINITION 3. The solutionV E Cimp[[-a,O] ^Uffi.+,ffi.] nC¹(Uk"=o(tk,

tk+l),lR) of the differential inequality(5), (6) is called eventually positive (negative), if there exists a numbert* ~ 0such that V (t) >0(V (t) < 0) for t ~ to.

THEOREM 1. Let tile lollowing conditions hold:

1. AssumptionsH1-H6 are fulfilled.

2. g(tk,x,~) :::::; Lk~, k = 1,2, ... ,x E n,~ Effi.+, Lk^{~ 0areconstants} andg(tk,X,~)= -g(tk,x,-~).

3. The differential inequality(5), (6) and the differential inequality (10) V'(t) +P(t)f(V(t - a)) :::::; -Ho(t), t =Jtk,

(11) V(tk) :::::; ⁽¹+Lk)V(t;;), have no eventually positive solutions.

Then each nonzero solution u E C²(E \ Eimp)n C¹(E* \ Eimp) of problem (1)-(3) oscillates in the domain E.

Proof. Suppose the conclusion of the theorem is not true, i.e., u(t,x) is a nonzero solution of the problem (1 )-(3) which is of the classC²(E \ Eimp )nC¹(E* \ Eimp), and it has a constant sign in the domain Ep. = [J.L, +(0) x n, ^fL ~ O. Ifu(t, x) > 0 for (t, x) ^E Ep., then it follows from Lemma 1 that V(t) is a positive solution of the differential inequality (5), (6) for t _~ fL+a, which contradicts condition 3 of the theorem. If u(t, x) < 0 for (t, x) E El" then the function -u(t, x) is a solution of the problem

Ut(t, x) - a(t).6.u(t, x) +p(t, x)f(u(t - a, x))

= -H(t, x), (t, x) E E \ Eimp,

u(t,x) -u(C,x) =g(t,x,u(C,x)), (t,x) E E;mp,

auan(t, x)+i(t, x)u(t, x) = 0, (t, x) ^E (ffi.+ \ Jimp )^X an,

which is positive inEfl." From Lemma 1 it follows that

!1-u(t,X)]dX ( [dx) -1

is a positive solution of the differential inequality (10), (11) fort ~ J.L+a which also contradicts condition 3 of the theorem. 0

THEOREM 2. Let the following conditions hold:

1. PE Cimp[lR+,lR-i-], Ho E Cimp[lR+,lR].

2. f(u) 2::0 foru 2:: O.

00

3.

L

k=l

4. For any number io2::a we have

t

liminf!_t-oo

IT

to ^S<tk5,t

Then the differential inequality (5), (6) has no eventually positive solutions.

Proof. Suppose the conclusion of the theorem is not true and letV(t) be a positive solution of the differential inequality (5), (6) in the interval [t*,+00), t* 2:: O. Then it follows from conditions 1 and 2 of the theorem that

V'(t) :::; Ho(t), t 2::t*+a, t i= tk.

Integrating over the interval lib t], t*+a:::;i1<t, we obtain

t

(12) V(t):::;

IT

J ^IT

t1<tk<t_- t"₁ S<tk5,t

Conditions3and4of the theorem imply that the right-hand side of(12) is not bounded from below and henceV(t) cannot be eventually positive

solution. 0

COROLLARY 1. Let the following conditions hold:

1. AssumptionsH1-H6are fulfilled.

2. g(tk' x,€) :::; Lk€, k =1,2, ... ,i En, €ElR+, Lk 2:: 0areconstants

00

such that

L

k=1

3. For any number to 2::a we have

t

liminf!_t-oo

IT

to ^S<tk5,t

and

Forced oscillations of solutions 887

(13)

t

limsupj

IT

t-->oo to S<tk5,t

Tiwneach nbilZerosolutionu E: C~(E \ Eimp )^{f1 C"} lE" \ Eimp) of tile problem (1)-(3) oscillates in the domain E.

Corollary 1 follows from Theorem 1 and Theorem 2.

Now we give sufficient conditions for oscillation of the solutions of problem (1), (2), (4). Consider the following Dirichlet problem

~r.p+ar.p=⁰ in n,

r.ploo = 0,

wherea =const.It isknown that the smallest eigenvalueao of the prob- lem (13) is positive and the corresponding eigenfunction r.po(x) > 0 for x E n. Without loss of generality we may assume that r.po is normalized, i.e.,

!

n

Introduce the notations:

W(t) =

!

H1(t) =

j

n

LEMMA 2. Let the following conditions hold:

1. Assumptions H1-H3, H5, and H6 are fulfilled.

2.u E C2(E\Eimp)nCl(E*\Eimp) is a positive solution" of the problem (1), (2), (4) in the domain E.

3.g(tk,x,~) ~ Lk~, k = 1,2, ... ,x E n,~ ElR+, Lk ~0 areconstants.

Then the function W(t) satisfiesfor t ~ (T the impulsive ditIerential inequality

(14) W'(t) +o:oa(t)W(t) +P(t)f(W(t - (T)) ~ H1(t), t=I tk, (15) W(tk) ~ (1+Lk)W(tk").

Proof Let t 2 a. We multiply the both sides of equation (1) by the eigenfunction 'Po (x) and integrating with respect to x overn, we obtain

~

J

J

n n

(16) +

f

f

n n

From the Green formula it follows that

J

J

n n

(17) =-aD

J

whereaD >0 is the smallest eigenvalue of the problem (13).

Moreover, from the Jensen inequality

f

f

n n

(18) 2:P(')!

(I

~

Making use of (17) and (18), we obtain from (16) that

W'(t) +aoa(t)W(t)+P(t)f(W(t - 0")) :::; H1(t), t -Itk·

For t= t_k we have that

W(tk) - W(t;;) :::; L k

J

that is,

o

Analogously to Theorem 1, we can prove the following theorem.

THEOREM 3. Let the following conditions hold:

1. AssumptionsHI-H3, H5,and H6 are fulfilled.

2. g(tk' x,.;) :::; L k';, k =1,2, ... ,x E n, .;^E'lR+, _{L k}2 0are constants and g(tk' x,.;) = -g(tk' x, -';).

Forced oscillations of solutions 889 3. The differential inequality(14), (15) and the differential inequality

W'(t) +^Qoa(t)W(t) +P(t)f(W(t - a)) ~ -HI(t), t:/: tk, W(tk) _~ (1+^Lk)W(t_{k ),}

~a-,-e no 1!)\'~~llttll~JJy J)(,:;it:iV"f~solutions.

Then each nonzero solution u E C²(E \ Eimp)n CI(E* \ E;mp) of problem (1), (2), (4) oscillates in the domain E.

THEOREM 4. Let the following conditions hold:

1. a, PE Cimp[JR+,JR+], HI E Cimp[JR+,JR].

2. f(u) 2: 0 foru 2: O.

00

3.

L

k=1

4. For anynumber to 2: a we have

t

liminf!

IT

t~oo to S<tk'St

Then the differential inequality(14), (15) has no eventually positive solutions.

The proof of Theorem 4 is analogous to the proof of Theorem 2. Itis omitted here.

COROLLARY 2. Let the following conditions hold:

1. Assumptions HI-H3, H5, and H6 are fulfilled.

2. g(tk,x,~) ~ Lk~, k = 1,2, ... ,x En,~ EJR+, L k2:0are constants

00

such that

L

k=1

3. Foranynumber to 2: a we have

t

liminf!

IT

t~oo

to S<tk'St "

and

Then each nonzero solution U E C²(E \ Eimp)nC¹(E* \ Eimp) of problem (1), (2), (4) oscillates in the domain E.

Corollary 2 follows from Theorem 3 and Theorem 4.

ACKNOWLEDGEMENTS. The present investigation was supported by the Bulgarian Ministry of Education and Science under grant MM-702.

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Higher Medical Institute P. O. Box 45, Sofia-1504 Bulgaria