Nonlincor Ann/~vsis, Theory. Merho& & Apphcadonr, Vol. 12, No. 12, pp. 1377-1383. 1988.

Printed in Great Britain. 0362%546X/t% 13.0 + .OO

0 198R Pergamon Press plc

QUENCHING PHENOMENA FOR SINGULAR NONLINEAR PARABOLIC EQUATIONS

C. Y. CHAN

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

and

MAN KAM KWONG

Northern Illinois University, DeKaIb, IL 60115. U.S.A., and University of Southwestern Louisiana, Lafayette. LA 70504, U.S.A.

(Received 2 June 1987; received for publication 14 April 1988)

Key words andphrases: Quenching, singular nonlinear parabolic equations. global existence. differential inequalities.

1. INTRODUCTION

KAWARADA [l] introduced the concept of quenching through the study of the following first initial boundary value problem:

Lu = -(l - u)-’ in Q, (1.1)

u(x,O)=O for-aSxSa, (1.2)

~(-a, t) = 0 = ~(a, t) fort > 0, (1.3)

where Lu = u, - u,, and 52 = {(x, t): -a <x < a, 0 < t < T}. The equation (1.1) arises in the study of a polarization phenomenon in ionic conductors. The solution u is said to quench if there exists a finite time T such that

sup{]u(x,t)l: -u~x~u}--,~ ast+T-, (1.4)

where u denotes u,. A necessary condition for (1.4) to hold is

max{u(x, t) : -a S x S a}+ l- as t--, T-. (1.5)

Kawarada’s main result claimed that (1.5) implied (1.4), and hence the two conditions were equivalent. Thus in studying quenching, Walter [2], and Acker and Walter [3, 41 used the necessary condition. The latter two papers studied qualitatively the more general equations:

Lu = -f(u), (1.6)

Lu = -g(u, ux), (1.7)

subject respectively to (1.2) and (1.3). For each problem, they showed that there was a critical length 2u* such that if a < a*, then u existed for all t > 0, and if a > a*, then quenching occurred. Results on the behavior of the solution of the equation (1.6) subject to (1.2) and

1377

1378 C.Y. CHAN~II~ M.K. KWONG

(1.3) with a = a* were given by Levine and Montgomery [5]. For further references, we refer to the survey given by Levine [6]. Recently, Chan and Chen [7] gave a numerical method to determine the critical lengths with the use of the monotone method (cf. Ladde, Lakshmikantham and Vatsala [8]): numerical examples corresponding to various functions f(u) were also given.

There is a gap in Kawarada’s proof that (1.5) implies (1.4). He assumed that the constructed function I@@, t) satisfied the heat equation on the curves s(‘)(t) for r sufficiently close to T. Such an assumption is, in general, false; furthermore, it seems that it is difficult to extend his technique to a more general nonlinear forcing term than -(l - u)-‘.

The purpose here is to establish Kawarada’s claim for the singular nonlinear equations (1.6) and (1.7) subject respectively to (1.2) and (1.3). Our proofs are different from his, and we include his result as a special case. Besides the first boundary conditions (1.3), we also consider the second boundary conditions,

u,(-a, t) = c(t), u,(a, t) = -c(t), t > 0, (1.8)

where c is continuous and bounded below by a positive constant. In Section 2, we study the equation (1.6). In Section 3, we investigate the equation (1.7).

2. THE CASEf(u)

Let us consider the first initial boundary value problem for (1.6). The following result generalizes the main claim of Kawarada.

THEOREM 1. Under the hypotheses that

f(0) ’ 07 f E C’[O, b),

I

h f(u) du = x,

0

(2.1)

(2.2)

where the positive constant b is such that f(u) tends to infinity as u tends to b-, if ~(0, t) tends to b- as t tends to T-, then the solution u of the first initial boundary value problem (1.6), (1.2) and (1.3) quenches.

Proof. Since f is continuous for u E [0, b), there exists a positive constant k such that f(u) z= -k. From Nagumo’s lemma, u is unique (cf. Acker and Walter [3]). It follows from their theorem l(b) that for each fixed I > 0, u attains its maximum at x = 0, and is strictly decreasing for x E (0, u]. Suppose u(x, t) remains bounded as ~(0, t) tends to b-. Then, we can choose a positive constant M such that

u(x, t) < M - k, b < Mu2/2. (2.3)

Let p E (3b/4, b) such that f(u) > 2M for u E [p, 6). Let t be some arbitrarily fixed time close to the quenching time T such that ~(0, t) > p. We have

u,,(x, f) <M - k -f(+, 0). (2.4)

Let xi denote the first positive x-coordinate at which u(x,, t) = /3. On [0, xi], it follows from

Singular nonlinear parabolic equations

(2.4) and f(u) > 2M that u,, < -k -f(u)/2. Hence,

u*X < -f(u)/2.

1379

(2.5)

For each fixed r, and x E [0, a], U, c 0. Let C-Y denote ~(0, t). From (2.5),

u;(xt, r) z= - /X&x, r))~,(x, r) dw = j?(u) du. 0 B

Since j3 is fixed, and the last integral tends to infinity as Ed tends to b-, we have U&X,, t) tends to negative infinity as r tends to T.

Because 2M <f(u), it follows from (2.4) that u, < -M. Integrating this twice with respect

to x, we obtain

u(x, t) < ~(0, t) - Mx2/2.

In particular, p < b - Mx:/2. This with (2.3) gives

x1 < [2(b - @/Ml”’ ==c a/2.

Thus, the length of the interval [xi, a] is at least a/2. Since f(u) 2 -k, it follows from (2.4) that

4x, t) < M.

Integrating this twice over [xl, x], we obtain

u(x, t) < /3 + (x - xl)u,(xl, t) + M(x - x,)‘/2.

(2.6)

Hence,

~(a, r) < /3 + (a - xI)u,(xl, t)/2 + Ma2/2.

This implies that ~(a, t) tends to negative infinity as t tends to T, in contradiction to the fact

that ~(a, t) = 0. We remark that in the case f(0) = 0, u = 0 is the only solution. Our next result shows that

for each value of t > 0, u attains its maximum at x = 0.

THEOREM 2. If (2.1) holds, and f’ is nondecreasing, then for the first initial boundary value

problem (1.6), (1.2) and (1.3), u(-x, t) = u(x, t), and

max{o(x, t) : -a s x G a} = ~(0, t) for each fixed t > 0.

Proof. Since u is unique, u(x, t) = u( -x, t), which gives u(x, t) = u( -x, t). This in turn gives u,(O, t) = 0. Now, u satisfies Lu = -f’(u)u in 9, u(x, 0) = q(x) for -a G x c a, ~(-a, t) = 0 = ~(a, t) for 0 < t < T, where q is some nonnegative function, symmetric about the line x = 0 since u 2 0, which follows from theorem l(a) of Acker and Walter [3] that u is strictly increasing with respect to t for -a < x < u. Because f’(u) is bounded above, we may apply the strong

maximum principle (cf. Protter and Weinberger [9, pp. 173-1751) to conclude that u is positive in Q. Let us consider the function w(x, t) = u(x + h, t) - u(x - h, t) in the region

R = {(x, t) : -a + h <x < 0,O < t < T},

where 0 < h. Since f’(u) is nondecreasing, and u > 0, we have [L + f’(u(x + h, t))]w S 0 in

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R. Also,

C. Y. CHAN and M. K. KWONG

‘w(x, 0) = lim u(x + h, At) - u(x - h, At) > 0

- 7 Af+O+ At

w(-a + h, t) = ~(--a + 2h, t) > 0, w(0, t) = 0.

It follows from the strong maximum principle that w > 0 in R. Therefore.for a fixed t > 0, u is strictly increasing in x for -a < x < 0, and hence by symmetry, u is strictly decreasing in x for 0 < x < II. Thus, the theorem is proved.

It follows from theorems 1 and 2 that at quenching, ~(0, t) tends to infinity as t tends to T. The following results are for the second initial boundary value problem.

THEOREM 3. If (2.1) and (2.2) hold, then a solution u of the second initial boundary value problem (1.6)) (1.2) and (1.8) has the following properties.

(a) u is unique. (b) u(x, t) = u(-X, f); furthermore for each fixed t > 0, u is strictly increasing in x for

-UCXGO. (c) If ~(0, t) tends to b- as t tends to T, then u quenches.

Proof (a). Suppose there are two distinct solutions ui and u2. Let s = u, - u2. Then, [L + f’(r])]s = 0 in S2, s(x, 0) = 0, ~,(-a, t) = 0 = ~,.(a, t), where q lies between ul and u2. Since f’(q) is bounded, we can apply the strong maximum principle and the parabolic version of Hopf’s lemma (cf. Protter and Weinberger [9, pp. 174-1751) to arrive at the solution is unique.

(b) Because u is unique, we have u(x, f) = u(-x, t). Let z(x, f) 3 u(x + o, t) - u(x - u, f) in

D = {(x, t): -a + C-J < x < 0,o c f c T},

where 0 < u. Then, [L +f’(c)]z = 0 in D, z(x, 0) = 0, and ~(0, t) = 0, where ,!j lies between u(x + u, t) and u(x - cr, t). If z < 0 somewhere in D, then z attains its minimum somewhere at x = -a + cr, say at the point (-a + o, t). By choosing u sufficiently small, this contradicts ~,(-a, t) being bounded below by a positive constant. Therefore, z > 0 in D. If z = 0 somewhere in D, then by the strong maximum principle, this again contradicts the given boundary condition.

(c) To show that u quenches, we follow the proof of theorem 1 until (2.6), which we integrate from xi to x to obtain

n,(x, r) < U,(XI 1 z> + M(x - Xl).

This gives -c(r) < uX(xI, t) + M(u - xi), and we have a contradiction since uX(xi, t) tends to negative infinity as t tends to T.

3. THE CASE g(u, u,)

Acker and Walter [4] assumed that

g(u, p) tends to infinity uniformly

over all p in any bounded interval as u tends to b-. (3.1)

Singular nonlinear parabolic equations 1381

Therefore for every positive constant p, there exists a function VP(u) such that 1+9Ju) G g(u, p) whenever Ip] G p, and qP(u) tends to infinity as u tends to b-.

Let r be some arbitrarily fixed time close to T. From theorem l(b) of Acker and Walter [4] for the problem (1.7), (1.2) and (1.3), there exists a function q(t) such that for each t E [0, T), u is monotone increasing in x on [-a, 9((t)] an d monotone decreasing in x on [q(t), a], where -a < q(r) G a with u,(q(t), r) = 0. Since u(a, t) = 0, and u > Oin 52, the length of the interval [q(t), u] is greater than zero. We have the following quenching result.

THEOREM 4. Under the hypotheses that g E C’([O, b) x (- cc, m)), g(O,O) > 0, (3.1) holds,

I

b

v&)du = ~4,

0

and there exist two positive constants kr and k2 such that

-kzp2 - kr Gg(u,p), (3.2)

if U(Q)(~), t) tends to b- as t tends to T, then the solution u of the first initial boundary value problem (1.7), (1.2) and (1.3) quenches.

Proof. From the strong maximum principle, u is unique. Suppose u(x, t) remains bounded as u tends to b-. Then, we can choose a positive constant M such that

u(x, r) < M - k,, b < M[u - q(t)]*/2. (3.3)

Let p be arbitrarily large and /3 E (3b/4, b) such that r+!~Ju) > 2M for u E [/3, b). Also, let t be such that (Y > fi, where (Y denotes u(QJ(~), r). We have

u, < M’- kl - g(u, u,). (3.4)

Let x1 denote the first x-coordinate to the right of q(t) at which u(xr, t) = 6. On [q(t), x,], (3.4) gives

u, < M - kr - w,(u) < -kr - w,(u)/2.

We have

uX.X < - VJp (u)/2. (3.5)

Because u is monotone decreasing on [q(t), x,], u, 6 0. Multiplying (3.5) by u, and integrating from q(t) to x1, we have

Since /3 is fixed, and the integral on the right-hand side tends to infinity as (Y tends to b-, it follows that uz(xr, t) tends to infinity as t tends to T. Thus at some point x2 E [q(r), x,], uX(x2, r) < -p, and hence uz(x2, t) > p2.

Let x0 be the smallest value in the interval [q(t), x1] such that u,(xo, t) = -p. Then,

]u,(x, t)] s p for q(t) G x G x0.

1382 C.Y. CHAN and M. K. KWONG

This gives

g(u, u,) 5 VP(u) for q(t) <x C x0.

Since M < v,,(u)/2 for u E [p, b), it follows from (3.5) that u,(x, t) < -M. As in the proof of theorem 1, we can show that the length of the interval [x0, a] is at least [a - ~~(t)]/2.

For x E [x0, a], we have u E [0, u,(x,,, r)]. Let R denote uf(x, t). For a fixed t,

dR du = 2[u(x, r) - g(u, &)I,

From (3.2) and (3.3),

R(xo, t) = p2.

dR du < 2(k2R + M).

The solution of the initial value problem,

dS z=2(k,S+M), S(xo,t)=p2,

is given by

S = (p* + M/k,) exp{-2k2[u(x,, t) - u(x, t)]} - M/k*.

Since p can be chosen to be arbitrarily large, we may assume

Mr2 p*+pJ+F,

2 2

where r2 denotes exp[2k2u(x,,, t)]. Thus, S Z= r-*p2/4. From the theory of differential inequalit- ies, R 3 S. Hence,

&(X9 r) =5 -p/(2r) forxo < x C u.

Integrating from x0 to a, we have

~(6 r) c u(xo, r) - ~(a - x0)l(2r),

from which,

0 < b - ]a - v WldW. Since p can be chosen arbitrarily large, we have a contradiction.

We remark that in the case g(0, 0) = 0, u = 0 is the only solution. An argument similar to that of theorem 3 gives the following results.

THEOREM 5. Under the hypotheses of theorem 4, if g(u,p) = g(u, -p), then a solution u of the second initial boundary value problem (1.7), (1.2) and (1.8) has the following properties.

(a) u is unique. (b) u(x, t) = u(-x, t); furthermore for each fixed t > 0, u is strictly increasing in x for

-aGxcO. (c) If ~(0, t) tends to b- as t tends to T, then u quenches.

Singular nonlinear parabolic equations 1383

REFERENCES

1, KAWARADA H., On solutions of initial boundary problem for ~1, = u,, + l/(1 - u), Publns RIMS, Kyo~o Univ. 10, 729-736 (1975).

2. WALTER W., Parabolic differential equations with a singular nonlinear term, Funkcialaj Ekvacio 19, 271-277 (1976).

3. ACKER A. & WALTER W., The quenching problem for nonlinear parabolic differential equations. Lecfure Notes in Mathematics 564, 1-12, Springer, New York (1976).

4. ACKER A. & WALTER W., On the global existence of solutions of parabolic differential equations with a singular nonlinear term, Nonlinear Analysis 2, 499-505 (1978).

5. LEVINE H. A. & MONTGOMERY J. T., The quenching of solutions of some nonlinear parabolic equations, SIAM J. math. Analysis 11, 842-847 (1980).

6. LEVINE H. A., The phenomenon of quenching: a survey, in Trends in the Theory and Practice of Non-linear Analysis (Edited by V. LAKSHMIKANTHAM), pp. 275-286. North-Holland, New York (1985).

7. CHAN C. Y. & CHEN C. S., A nonlinear singular parabolic initial boundary value problem. in Nonlinear Analysis and Application (Edited by V. LAKSHMIKANTHAM), pp. 115-119. Marcel Dekker, New York (1987).

8. LADDE G. S., LAKSHMIKANTHAM V. & VATSALA A. S., Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman, Boston (1985).

9. PROITER M. H. & WEINBERGER H. F., Maximum Principles in Differential Equations. Prentice-Hall, Englewood Cliffs, NJ (1967).