Pergamon

Nonlinear Analysis, Theory, Methods & Applications, Vol. 22, No. 1 I, pp. 1323-1328, 1994 Copyright 0 1994 Elsevier Science Ltd

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0362-546X/94 $7.00+ .oO

IMPULSIVE QUENCHING FOR REACTION-DIFFUSION EQUATIONS

C. Y. CHAN, L. KE and A. S. VATSALA

University of Southwestern Louisiana, Lafayette, LA 70504-1010, U.S.A.

(Received 4 March 1993; received for publication 16 June 1993)

Key words and phrases: Impulses, quenching, critical length, maximum principle, lower solution, upper solution.

1. INTRODUCTION

OWING to short-term perturbations, many evolution processes at certain moments of time experience changes of state abruptly. Since the durations of the perturbations are negligible in comparison with the duration of each process, it is natural to assume that these perturbations act instantaneously in the form of impulses (cf. [l]). Since the introduction of the concept of quenching in 1975, many scientists have studied initial-boundary value problems for hyperbolic and parabolic equations. More recently, Chan and Ke [2] have extended the concept of critical lengths to periodic solutions of semilinear parabolic systems. For more recent references on quenching, we refer to the work of Chan [3].

Let

and a, T and CJ be positive constants. We consider the following quenching problem with impulses: for n = 1,2, 3, . ..,

Hu = -f(u), O<x<a, (n- l)T<tsnT-, (1.1)

u(x, 0) = 0, Olxsa, (1.2)

u(x, nT) = m(x, nT-), Olxsa, (1.3)

~(0, t) = 0 = u(a, t), 0 < t, (1.4)

where for some positive constant c, f E C’[O, c) is positive, f’ 2 0, and lim,,,- f (u) = 00. This represents a reaction-diffusion phenomenon subject to zero initial-boundary data; after each T-interval of time, an impulse (1.3) is given to the system. We exclude the case o = 1 since this corresponds to a problem without any impulse.

The length a* is called the critical length of problem (1 . l)-( 1.4) if for a < a*, the problem has a unique global solution, and if for a > a*, no global solution exists.

Let Q = (0, a) x (0, T-1, and fi be its closure. By the transformation,

u’“‘(x, 1 - (n - l)T) = u(x, t), (n - l)T < t I nT, n = 1, 2, 3, . . . ,

the system (1. I)-( 1.4) becomes

uyx t) = 0

Hu(“) : -f@‘“‘) on fi,

in a,

1323

(1.5)

(1.6)

1324 C. Y. CHAN et al.

u’“‘(x, 0) = cd-‘)(x T-) 7 3 Osxsa, (1.7)

~‘“‘(0, t) = 0 = d”‘(a, t), 0 < t I T-, (1.8)

wheren= 1,2,3 ,.... Thus, global existence of a solution of problem (1 . l)-( 1.4) is equivalent to existence of 2P for all positive integers n.

Here, we would like to study the effect of impulses on the critical lengths.

2. THE CASE o < 1

Let us consider the singular two-point boundary value problem,

U”(X) = -f( U(X)), U(0) = 0 = U(a). (2.1)

Since a solution U is symmetric with respect to the line x = a/2, it follows from Chan and Kaper [4] that problem (2.1) has a solution U satisfying 0 < U < c for 0 < x < a, provided a is sufficiently small. In terms of Green’s function corresponding to the operator d2/dX2 subject to first boundary conditions,

U(x) = y s xu(WO) dC + t a(a - Of(W)) dt. 0 i’ x

By direct differentiation, CJ E C2[0, a].

LEMMA 1. If problem (2.1) has a solution U E C2[0, a] such that U(x) > 0 for 0 < x < a, then fora< 1,

U > u@) in Q, II= 1,2,3 ,.... (2.2)

Furthermore, problem (1.5)-( 1.8) has a unique solution [u’“‘):, 1 satisfying

~(~1 < u(“+i) in Q, n = 1,2,3, . . . . (2.3)

Proof. By the strong maximum principle, u(l) > 0 in a. It follows by mathematical induction that u(“) > 0 in Q for all positive integers n.

By the strong maximum principle, U > u (l) in Q Let us suppose (2.2) is true for some posi- . tive integer k. Let w = U - u (kc1) Then for some r between U and u@+‘), .

(H + f’(O)w = 0 in Q

w(x , 0) > (1 - a)#‘(~ 9 T) > 0,

w(0, t) = 0 = w(a, t), 0 < t s T-.

By the strong maximum principle, w > 0 in Q, and, hence, (2.2) is true. A similar argument shows that (2.3) is true. Since 0 is a lower solution and U E C2[0, a] is an upper solution, it follows from theorem

4.2.2 of Ladde et al. [5, pp. 1431 that problem (1.5)-(1.8) has a solution which is unique by the maximum principle.

Let u@)(x, t; a) denote a solution of problem (1.5)-(1.8), and Sz, = (0, al) x (0, T].

Impulsive quenching

LEMMA 2. If a, < a2, then for all positive numbers (T,

zP(x t. al) < 9 9 zP(x t. ~7~) in Q,, 9 3 n = 1,2,3 ,...,

1325

(2.4)

provided the solutions exist.

Proof. Let

w = U”‘(X 1’ a,) - 9 9 U”‘(X 1’ a ) 3 9 ,’

Then for some r between u”‘(x, I; a,) and u(‘)(x, t; a,),

w + f’K))w = 0 in CJr,

w(x, 0) = 0, 0 I x I a,,

w(0, t) = 0, W(lzl) t) = u(‘)(a,, t; UJ > 0, 0 < t I T-.

By the strong maximum principle, w > 0 in Sz, . The lemma then follows by using mathematical induction.

We remark that starting with n = 2, inequality (2.3) also holds for (x, t) E (0, a) x (01, and inequality (2.4) also holds for (x, t) E (0, a,) x {O).

We now give a criterion for existence of a unique critical length for o < 1.

THEOREM 3. If T > (1 - a)c/f(O) when o < 1, then problem (1. I)-(1.4) has a unique critical length.

Proof. By lemma 1, problem (1 .5)-( 1.8) has a unique solution (u’“‘]~= 1 provided a is suffi- ciently small. When problem (1.5)-(1.8) has a unique solution (u’“‘(x, t; a));= L for some a, say (Y, it follows from lemma 2 that the problem with a < CY has a solution and this solution is unique by lemma 1. Thus, if the theorem is false, then given any length a, problem (1.5)-(1.8) has a solution.

Let

U@)(X) = L

i’

T

T, u(“)(x t) dt

’ ’ (2.5)

Then, u@) < c since u(n) < c in Sz. Integrating (1.6) with respect to t and using (2.5), we have fern= 1,2,3 ,...,

rjn)n(x) = ; [~‘“‘(x, T) - o.u(“-‘) (x T)] - !_ ‘f(u’“‘(x t)) dt , .i T, ’ .

Let lim,,, u(“) = g. Since u@) < c, we have _u I c. Let

E = )[f(O)T - (1 - a)~].

Then, E > 0 by assumption. It follows that there exists a positive number N such that for n 2 N,

u(“)(x, T) - cm (n-‘)(~, T) < (1 - a)c + E,

1326 C. Y. CHAN et al.

and, hence, for n 2 N,

zPU(x) < ; [(l - o)c + & - f(O)T] = -; .

Since ~‘“‘(0) = 0 = v’“‘(a), it follows from the strong maximum principle that

U@)(X) > Ex(a - x) 2T -

This gives

max U(“)(X) 2 -5 a2 OSXSU 8T ’

which implies that maxO c X 5 (I U@)(X) > c for sufficiently large a. This contradicts v(‘) < c, and, hence, the theorem is proved.

3. THE CASE o > 1

For any positive constant b < c, let 6 = (c - 6)/2, and M = f(b + 6). We consider the following impulsive problem: for n = 1,2,3, . . . ,

pyx, t) = 0 on Q

Hz’“’ = -M in Q

z@)(x , 0) = a$- “(x T-) 3 , Osxsa,

~‘“‘(0 9 t) = 0 = z(“)(a t) , 9 0 < t I Tp.

LEMMA 4. If o > 1, then for any given T (> 0), there exists a length a such that

z@) < b + 6 in Q, n = 1,2,3 ,..., (3.1)

z@)(x 9 T) < b a’

Osxla, n = 1,2,3, . . . . (3.2)

Proof. It follows from theorem A.4.1 of Ladde et al. [5, p. 2171 that for each n, z(“) exists and is unique. By direct calculation, w(x) = Mx(a - x)/2 is the solution of the problem,

w” = -&f 9 w(0) = 0 = w(a).

By the strong maximum principle, z(l) < w in a. Let a be chosen such that

It follows from w 5 Ma218 that

z(l) < 6 in Sz,

(3.3)

(3.4)

(3.5) z(l)(x 9 T) < b 2a ’

Osxla.

Impulsive quenching 1327

Let us assume that for some positive integer k, (3.1) and (3.2) are true. By the principle of superposition, i?+‘) = z(‘) + W where W is the solution of the problem,

HW=O in 0,

W(X, 0) = crz’k’(x, T), Osxsa,

W(0, t) = 0 = W(a, t), 0 < t 5 T-.

By the weak maximum principle, W < b in 0. By (3.4),

ZCk+‘) < b + 6 in Sz.

Now

W(x, T) = : s

a i 0 n=l

sinFxsinF[exp WI - s 2T zCk)([,T)dl

< 2bjI ew[ -r;;n>iT] < 2b l;exp[ -(;yTt2] dr

ab =-

Y!X’

By (3.3) and (3.5), we have zCk+‘) (x, T) < b/a for 0 I x I a. The lemma then follows by the principle of mathematical induction.

LEMMA 5. Under the hypotheses of lemma 4, problem (1.5)-(1.8) has a unique solution satisfying

u(@ I z@) in Sz, n = 1,2,3 ,.... (3.6)

Proof. Since z@) -C b + 6, we have

Hz(“) = -M 5 -f(z’“‘)_

Thus, for some r between zCn) and u(“),

(H + f ‘([))(z’“’ - u(“)) I 0 in CJ.

By the strong maximum principle, (3.6) is true for n = 1. Since

z@)(x , 0) - ucn)(x , 0) = a[$“-‘1(x , T) - #-‘) t-c TN 9

(3.6) follows by using mathematical induction. For each n, 0 and z@) are, respectively, the lower and upper solutions of u @). It follows that problem (1.5)-(1.8) has a unique solution.

Now, we are ready to study existence of a unique critical length when o > 1.

THEOREM 6. If (T > 1, then for any given T (>O), problem (1 .l)-(1.4) has a unique critical length.

1328 c. Y. CHAN et al.

Proof. Given any T, it follows from lemmas 4 and 5 that we can always choose the length a sufficiently small such that problem (1.5)-(1.8) has a unique solution. When it has a unique solution for some a, say (Y, it follows from lemma 2 that problem (1.5)-( 1.8) with a < CY has a solution, which is unique by lemma 5.

If the theorem is false, then given any length a, problem (1.5)-(1.8) has a solution, and, hence, problem (1 .l)-( 1.4) has a solution U. Let ii be the solution of problem (1 . l)-( 1.4) with 0 = 1. We denote its corresponding solution of problem (1.5)-(1.8) by (ti’“‘)~= 1. By the strong maximum principle, u(“) > U’“’ in Sz for n = 2, 3,4, . . . . Hence, u > zi for (x, t) E (0, a) x (0, co). It is known that u” reaches the value c somewhere in finite time when II is sufficiently large (cf. Chan and Kaper [4]). This contradiction proves the theorem.

Acknowledgement-The authors would like to thank Professor V. Lakshmikantham for the suggestion of incorporating impulses into quenching.

REFERENCES

1. LAKSHMIKANTHAM V., BAINOV D. D. & SIMEONOV P. S., Theory of Impulsive Differential Equations. World Scientific, Teaneck, New Jersey (1989).

2. CHAN C. Y. & KE L., Critical lengths for periodic solutions of semilinear parabolic systems, Dynam. Syst. Appl. 1, 3-11 (1992).

3. CHAN C. Y., New results in quenching, Proc. First World Congress of Nonlinear Analysts (Edited by V. LAKSHMIKANTHAM). Walter de Gruyter, Berlin (to appear).

4. CHAN C. Y. & KAPER H. G., Quenching for semilinear singular parabolic problems, SIAM J. math. Analysis 20, 558-566 (1989).

5. LADDE G. S., LAKsHMmANTnui V. & VATSALA A. S., Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman Advanced Publishing Program, Boston (1985).