On the stability of solutions of impulsive nonlinear parabolic equations

ESAIM: M

D ^RUMI B ^AINOV E ^MIL M ^INCHEV

On the stability of solutions of impulsive nonlinear parabolic equations

ESAIM: Modélisation mathématique et analyse numérique, tome 33, n^o2 (1999), p. 351-357

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Mathematical Modelling and Numerical Analysis M2AN, Vol. 33 N° 2, 1999, p. 351-357 Modélisation Mathématique et Analyse Numérique

ON THE STABILITY OF SOLUTIONS OF IMPULSIVE NONLINEAR PARABOLIC EQUATIONS

DRUMI B A I N O V1 AND E M I L M I N C H E V1

Abstract. Stability and asymptotic stability of the solutions of impulsive nonlinear parabolic équa- tions are studied via the method of differential inequalities.

Resumé. La stabilité et le comportement asymptotique de solutions d'équations paraboliques non- linéaires impulsives sont étudiés via la méthode des inégalités différentielles.

AMS Subject Classification. 35B35.

Received: June 2, 1997. Revised: January 5, 1998.

1. INTRODUCTION

The theory of the impulsive ordinary differential équations underwent rapid development in the recent years [1,3,7-9,11]. The impulsive differential équations are adequate apparatus for mathematical simulation of many pro cesses and phenomena in nature which are char act erized by the fact that the System parameters are subject to short-term perturbations in time.

However, taking more factors into account leads to the development of the theory of impulsive partial dif- ferential équations. This theory marked its beginning with the paper [10]. The impulsive PDE provide natural framework for mathematical simulation of numerous processes and phenomena in theoretical physics, popula- tion dynamics, bio-technologies, chemistry, impulse technique and économies. We would like to point out the applications of the impulsive PDE in the quantum mechanics. In 1992 it was introduced a model of impulsive moving mirror [13,14] presented by the apparatus of the impulsive PDE. For the present state of the theory of impulsive PDE we refer the reader to the monograph [2] and to the survey papers [4-6,12].

In the present paper we study the stability and the asymptotic stability of the solutions of impulsive nonlinear parabolic équations via the method of differential inequalities.

2. P R E L Ï M I N A R Y NOTES

Let ü C W1 be a bounded domain with a smooth boundary dfl and Q = ft U dû. Suppose that 0 = XQ < Xi < X2 < - - • < Xk < • • •

are given numbers such that lim^^oo ^fc = +oo.

Keywords and phrases. Stability of solutions, impulsive nonlinear parabolic équations.

1 Médical University of Sofia, P.O. Box 45, Sofia 1504, Bulgaria.

© EDP Sciences, SMAI 1999

352 D BAINOV AND E MINCHEV

Define

E = {(x,y) G M1+n : x ^ R+ ) y G 0 } , R+ = [0,+OG),

oo

,y)€E: xe (xk,xk+1), y G ft}, fc = o , i , . . . , r = | J rfc,

Let Cir^pf.E, R] be the class of all functions u: E -» R such that:

(i) The functions u |r U B , k = 0 , 1 , . . . , are continuous.

(ii) For each fc,fc = l , 2 , . . . , x = ccfc, there exists the limit

lim u(s,q)=u(x-,y), y G fi.

(5,gr)^(x,î/) s<as

(iii) For each fc, A; = 0 , 1 , . . . , x — xk, there exists the limit

lim u(s,q) = u(x+,y), y eU

and u(x, y) = u(x+, y), y G VL.

Dénote by M[n] the class of ail matrices A = [^]i<ij<n with real entries. Let ƒ : F x R x W1 x M[n]

y? : M+ x 9ÎÎ ->- M, ti0 : O ->- M, ^ : {xk}%xL1 x H x M ->> M be given functions.

Consider the initial-boundary value problem (IBVP):

y)), (x,y) G T, (1) (2)

u(x,y) = <^(x,y), ( x , y ) e E+x ÔQ, (3)

u{xk,y) = u(x-,y) + g(xk,y,u(x-yy)), y G ÎT, fc = 1, 2 , . . . , (4) (a;,^),... ⁾%ⁿ(a;⁾î/)), u^yy(x,y) = [^^y,(x,y)]i<^{Î J}<n.

Définition 1. A function u : S —>• M is said to be a solution of the IBVP (1-4) if:

(i) u G Cimp[i£,R], there exist continuous partial derivatives ux(x,y), uy(x,y), uyy(x1y) for (x,y) G F and u satisfies (1) on F.

(ii) u satisfies (2=4).

Définition 2. The function ƒ : F x R x Rn x M[n] -> R is said to be elliptic at F if for each point (x,y) G F and any Q7 S G M[n] the quadratic form

(Qî3 - S%J)XtX3 < 0

for arbitrary vector À G Rn implies where (u,P) G R x R™.

We adopt the following définitions of stability:

Définition 3. The trivial solution u(x,y) = 0, (x7y) G E of the IBVP (1-4) is said to be stable if for every e > 0 there exists J = S(e) > 0 such that |uo(y)| < S, y £ Q and |</?(a;,y)| < S, (as,y) G R+ x <9£3 imply

(:r,y)| < e on £*.

IMPULSIVE NONLINEAR PARABOLIC EQUATIONS 353 Définition 4. The trivial solution u(x,y) ~ 0, (x,y) G E of the IBVP (1-4) is said to be asymptotically stable if:

(i) It is stable.

(ii) There exists a positive number öo such that to every e > 0 there corresponds X(e) such that |uo(y)| < ÔQ, y sft and \y>(x,y)\ < SQ, (x,y) G M+ x dû imply |n(x,y)| < s ior x > X(e), y e f i .

3. MAIN RESULTS Theorem 1. Let the following conditions hold:

1. The function ƒ zs elliptic^at F.

2. T/iere eariste a function ƒ G C((E+ \ {x^}^!) x M, M) sucft iftoi

/(ar,î/,p,0,0)</(x,p) (5) /or(:z,s/)Gr, p e ! L

3. T/iere ea;ist5 a function g G C({xfc}g?=1) x M, M) swc/i ifta*

) (6)

;

4. T/te function p + g(xk,p) is nondecreasing on R for each k, k = 1, 2 , . . . 5. T/iere ea:zsis a function j(x) which is a maximal solution of the problem

7'0*0 = f(x,j(x)), x^xk,

7(0) = 70, (7) j(xk) = i(x^)+g(xk,<y(x^)), k = 1 , 2 , . . . ,

<7o, y e H , (8)

), (a, y) G l + x Ô£I (9) T/ien /or am/ solution u of the IBVP (1-4) tue fowe that

u{x,y) <7(x) on S. (10) Proo/. Let To > 0, ETQ = [0, To] x Çt and there exist a positive integer m such that xm <TQ < xm + i. We prove that

u{x,y) < 7(x) on ETo- (11)

There exists eo > 0 such that for 0 < e < £o5 a solution 7(-; e) of the problem

(12) fe = 1,2,... ,m,

354 D. BAINOV AND E. MINCHEV

is defined on [0,TQ] and lime_^o 7(#; £) = 70e)» uniformly on [O,To]. We prove that

u(x,y) < j(x;e) on ETQ. (13)

Suppose that (13) is not true. Then the set Z = {x G [O,Tb] : there exists y £ Q, such that u(x,y) > "Y(X;e)} is non-empty. Defining x = inf Z. It follows from (8) that x > 0 and there exists a point y <E Ü such that

u(x, y) < <y(x; e), (x, y) e [0, x) x Q, u(x,y) =<y(x;e).

There are three cases to be distinguished:

Case 1. y € ö^L Then in view of (9)

and we get a contradiction.

Case 2. (£,y) G F. Then we have

u(x,y) - 7(x;

, 5) > 7;(x; e), uy(a;,g) = 0,

Prom the ellipticity of ƒ and (5) we obtain

0 < ux(x,y) - 7;( x ; e )

< f(x, y, w(ï, y)? 0, 0) - ƒ(£, 7(ï; e)) - e < 0, which is a contradiction.

Case 3. x ~ Xk for some fc, 1 < k < m. Then we obtain from (14) that u{xl,y) 7(a^;e),

(15) u(xkiy) = {)

Prom (6), (15) and condition 4 of the theorem we conclude that 0-= u{xkiy) -i(xk\e)

= u>(Xk,y)+g(xk,y,u(Xk,y)) - l(x~\s) - g(xkyj(x^;e)) -e

< u{xl,y) +g(xk,u(Xk ,y)) -j(x^;e) - ^ ( ^ ^ ( ^ j e ) ) - e < 0, which is a contradiction.

Hence the set Z is empty and (13) follows. Since lime_j.o j(x;e) = 7(0?) uniformly on [0, To] we conclude that u(x,y)<y{x) on ETo.

Since TQ > 0 was arbitrary we get the estimate (10). D

IMPULSIVE NONLINEAR PARABOLIC EQUATIONS 355 Theorem 2. Let the following conditions hold:

1. Conditions 1-4 of Theorem 1 are fulfilled.

2. Tkere exists a function -y(x) which is a maximal solution of the problem (7), where

|«o(y)l<7o, yen, (16)

^(z)2/)|<7(z), OcsrtGR+xôfi. (17)

3. f{x,y,-p,-q,-r) = -f{x,y,p,^,r) for (x,y,p,q,r) eTxRxW¹ x M[n}.

4. g(x^k,y,-p) = -g(x^k,y,p), y G O, p G R, A; = 1,2,...

Then for any solution u of the IBVP (1-4) we have that

\^u(^xiV)\ < l(^x) ^on E-

Proof The estimate

u(a;,y)<7(ar) on E (18) follows from Theorem 1. Now we will prove that

-u{x,y) <j(x) on E. (19) ït follows from conditions 3 and 4 of Theorem 2 that the function —u(x, y) is a solution of the problem

v>x(x,y) = f(x,y,u(x,y)}Uy(x,y),Uyy(x,y)), (x,y) G T,

«(o,?/) = -uo(2/), y e n ,

u(a;, y) = - y?(a;, y), (x, y) G M⁺ x Ôfi,

u(xk,y) = u(x~,y) + g(xk,y,u(x~,y)), y G H , /c = 1 , 2 , . . .

Having in mind the estimâtes (16) and (17), we conclude by Theorem 1 that the inequality (19) holds true.

Relations (18) and (19) prove the conclusion of the theorem. • Theorem 3. Let the following conditions hold:

1. Conditions 1, 3, 4 of Theorem 2 are fulfilled.

2. The maximal solution *y(x) of the problem (7) is defined on M+ and it is nonnegative.

3. /(a,y,0,0,0) - 0 for (x,y) G I \ g(x^k,y,Q) = 0, y G Q, / ( z , 0 ) = 0, x G R+ \ { a : * } ^ , ff(x^fc,0) = 0, TTien t/ie stability or asymptotic stability of the trivial solution of the problem (7) implies stability or asymp- totic stability of the trivial solution of the IBVP (1-4).

Proof. Suppose that the trivial solution of the problem (7) is stable. Then for every e > 0 there exists S — 5(e) > 0 such that jo < Ô implies j(x) < e on M⁺. Then by Theorem 2 it follows that

\u(x,y)\ < j(x) on E provided that

\uo(y)\ <7o, y G H

356 D. BAINOV AND E. MINCHEV

and

\<p(x,y)\ <l(x), (x,y) 6R+XÔÎÎ.

This proves the stability of the trivial solution of the IBVP (1-4).

Analogously it can be proved that the asymptotic stability of the trivial solution of (7) implies asymptotic stability of the trivial solution of the IBVP (1-4). D

4. APPLICATIONS IN THE POPULATION DYNAMICS

Particular interest for the mathematical biology is the special case of IBVP (1-4) when f{x>y,p,q,r) = KT + p(a — bp²), K > 0, a > 0, b> 0 are constants and ip(x, y) = 0. Then the IBVP (1-4) takes on the form

cyy) + u(x1y)(a~bu (s,y)), (z,y)Gl\ (20) yeH, (21) u(x, y) = 0, (x, y) e M+ x 0 « , (22) u(x^k,y) = u(x^,y) + g(x^k,y,u{x~^,y)), y G H, fe = 1 , 2 , . . . (23) The IBVP (20-23) describes a single species population in a bounded environment. The function u(xyy) represents the population density at the point y € Q, and time x > 0. Condition (23) describes instantaneous changes in the population density due to phenomena as: har vesting, disasters, immigration, émigration, etc.

Suppose that g(xk,y,p) = f3kp, pk > - 1 , y e H, k = 1, 2 , . . . , and

> 7>-0(*-*)

where L > 0, ƒ? > 0 are constants. Let 70 = maxî/€^no(y) and u b e a solution of the IBVP (20-23). Then we consider the problem

i(x) = 7(x)(a - bj2(x)), x £ xk, (24) 7(0) - 70, (25) l{x^k) = ⁷(xfc ) + & 7 K ) , fc = 1,2,... (26)

We substitute p(x) = ⁿ, ^N and obtain the problem 7 W

po = -5, 7o

= 1, 2,.

IMPULSIVE NONLINEAR PARABOLIC EQUATIONS 357

Then we have

x

f

Therefore

7 ( ) - L U

and by Theorem 2 we conclude that

O n t h e other h a n d , we have by T h e o r e m 3 t h a t t h e stability of t h e trivial solution of (24-26) implies stability of t h e trivial solution of I B V P (20-23).

The authors express their deep gratitude to the référée for his valuable advices and helpful suggestions. The present investigation was partially supported by the Bulgarian Ministry of Education and Science under grant MM-702.

R E F E R E N C E S

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[10] L.H. Erbe, H.L Preedman, X.Z. Liu and J.H. Wu, Comparison principles for impulsive parabolic équations with applications to models of single species growth. J. Austral. Math. Soc, Ser. B 32 (1991) 382-400.

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Appl. Anal. 1 (1997) 351-369.

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