quenching for coupled semilinear reaction-diffusion problems

Nonlrneor Analysis, Theory, Methods & Applications, Vol. 21, No. 2, pp. 143-152, 1993 Punted in Great Britain.

0362-546X193 $6.W+ SKI 0 1993 Pergamon Press Lfd

QUENCHING FOR COUPLED SEMILINEAR REACTION-DIFFUSION PROBLEMS

C. Y. CHANT and DAVID T. FUNGI

TDepartment of Mathematics, University of Southwestern Louisiana, Lafayette, LA 70504, U.S.A.; and $Department of Mathematics, Southeastern Louisiana University, Hammond, LA 70402, U.S.A.

(Received 2 January 1992; received in revised form 15 October 1992; received for publication 26 January 1993)

Key words and phrases: Coupled system, reaction-diffusion, critical length, blow-up of the time derivative.

1. INTRODUCTION

LET

i-2 = (094 x (0, v, 82 = w, 4 x 101) u (IO, 4 x (0, T)),

where T I co. We call aM the parabolic boundary of Q. Acker and Walter [l, 21 showed that a unique critical length a* exists for each of the following problems

Hp = -f(p) in Q fi = 0 on aa, (1.1)

H, = -g(fi,& in Q2, ,u = 0 on an, (1.2)

where f(p) tends to infinity as P approaches some positive constant c, and g(p, v) tends to infinity uniformly over all v in any bounded interval as P approaches c. Here, the critical length a* is the length such that the solution exists for all time t > 0 when a < a*, and when a > a*,

max(p(x, t): 0 5 x 5 a) + c- asC-+ T- (1.3)

for some finite time T. In the latter case, ,D is said to quench (in accordance with Acker and Walter [l]), and Tis called the quenching time. Under weaker hypotheses, Chan and Kwong [3] extended the above result for the problem (1.2) to the problem,

HP = -h(x, ,uu, PJ in Q, p = 0 on an.

Behavior of the solution of problem (1 .l) with a = a* was investigated by Levine and Montgomery [4]. Let L = H + bx-’ Wax, where b is a constant less than 1. Existence of a* and its determination by computational methods were given by:

(i) Chan and Chen [5] for the problem,

L,u = -(l - ,D-’ in 0, ,u = 0 on an;

(ii) Chan and Kaper [6] for the problem,

Lp = -f(,u) in a, ,D(x, 0) = 0 for 0 5 x 5 a, fi(O, t) = 0 = ,~~(a, t) for t > 0; (1.4)

(iii) Chan and Cobb [7] for the problem,

Lp = -f(p) in 0, ~(x, 0) = 0 for 0 5 x 5 a, ~(0, t) = 0 = ~~(a, t) + @(a, t) for t > 0;

143

144 C. Y. C&w and D. T. FUNG

where k is a positive constant. So far, for a coupled parabolic system, only the critical length a* for global existence of the solution of a system of two equations has been studied by Chan and Chen [8]; no result has been given showing that for Q > a*, quenching occurs.

Kawarada [9] claimed that for problem (1.1) with f(u) = (1 - p)-‘, (1.3) implied

sup(I,&(x, t)l: 0 5 x 5 a) + 00 as t + T-. (1.5)

Chan and Kwong [lo] pointed out the main error in his proof, and established his claim for the more general problems (1.1) and (1.2) by using different methods. Chan and Kaper [6] showed that (1.3) implied (1.5) for problem (1.4). Acker and Kawohl [l l] proved Kawarada’s claim for the radially symmetric solution of the multidimensional semilinear heat equation, subject to the zero boundary condition and a nonnegative, radially symmetric and nonincreasing initial condition; such a solution satisfies the singular equation, Ly = -f(p) in Q, but with b a positive integer and ~~(0, t) = 0 for 0 < t < T. Using an argument based on modifications of Friedman and McLeod [12] for blow-up problems, Deng and Levine [ 131 established Kawarada’s claim for a multidimensional semilinear heat equation in a bounded convex domain with a smooth boundary. So far, all work on the blow-up of the time derivative pUt has been done on problems, each of which involves a scalar parabolic equation.

Here, we would like to study the coupled reaction-diffusion problem,

Hu = -f(u) in 0, u = 0 on ac2, (1.6)

where u and f(u) denote, respectively, the transpose of the vectors (u,, u2) and

(fi(UlY u,), fi(ul, u,)). For brevity, we use throughout this paper the convention that the subscript i denotes the values 1 and 2, respectively. We impose the following conditions on f:

(i) f(0) > 0; (ii) there exists a positive constant column vector q with q, 1 q2 such that f(w) is of class C2

in (-00, qt) x (-co, q2), and limW2+,,fi(w) = 00; (iii) to any E with 0 < E < q2, there exists a constant C such that

fdw) - f,(w) 2 C(W - WI)

for wi 5 qi - E;

(iv) the first and second partial derivatives of f(w) are nonnegative, and the first partial derivatives are bounded for wi 5 qi - E.

We remark that condition (iii) is used to show that u, 5 u2. With this, f(w) in condition (ii) can be of class C” in [0, q2) x [0, q2) since such a function f(w) with f(0) > 0 and nonnegative first partial derivatives is positive, and can be extended as a positive continuous function in (-co, q2) x (-00, q2); with this extended function, it follows from the strong maximum principle (cf. Protter and Weinberger [14, pp. 168-1691) that u > 0 in R. With u > 0 in Sz, the second assumption about boundedness in condition (iv) is always satisfied, and, hence, can be omitted. We also remark that it will be seen later that the blow-up of the time derivative of the solution does not depend on ut I Z.Q and q1 L q2.

A solution u of problem (1.6) is said to quench if

max(u,(x, t): 0 5 x 5 a) -+ q2 as t’T_

for some finite time T. As in a previous study, we assume a solution exists before its quenching time. In Section 2, we show that there is a unique critical length for global existence of the

Reaction-diffusion problems 145

solution and for quenching; it is determined by its corresponding steady-state solution. In Section 3, we established the blow-up of the time derivative of the solution. For illustration, an example is given.

2. CRITICAL LENGTH

The following theorem follows from lemmas 1 and 2, and theorems 3 and 4 of Chan and Chen [8].

THEOREM 1. (a) There exists at most one solution u of problem (1.6); (b) the solution u > 0 in Q; (c) the solution u is strictly increasing in t for 0 < x < a; (d) in Q ur 5 u,; (e) if ui I qi - E, then the steady-state problem,

U” = -f(U) for 0 < x < a, U(0) = 0 = U(a),

has a solution, to which u converges componentwise uniformly from below as t tends to infinity;

(f) let u(x, t; a) denote a solution of problem (1.6). For any positive number q, u(x, t; a) < u(x, t; a + q) in Q2; furthermore,

uj(x, t; a) < min(ui(x + (3, t; a + rf): 0 5 D 5 q) in Q.

The next result shows that for each t, the absolute maximum of u occurs at x = a/2.

LEMMA 2. The solution u is symmetric about x = a/2. For each t E (0, T), the solution u is a strictly increasing function of x E (0, a/2) and a strictly decreasing function of x E (a/2, a).

Proof. Since u(a - x, t) satisfies problem (1.6), it follows from theorem 1 (a) that u is

symmetric about x = a/2. For any positive constant h (<a/2), let

v(x, t) = u(x + h, t) - u(x - h, t),

where (x, t) E (h, a - h) x (0, T). Then,

W(x, t) + c ’ afiieiY 2i) u, = o

j= 1 dUj ’ ’

where (ei, Ai) is a point on the line segment joining u(x + h, t) and u(x - h, t). Also,

v(x, 0) = 0 for h 5 x I a - h,

v(h, t) = u(2h, t) > 0 for 0 < t < T.

By the symmetry of u about x = a/2,

v(;,t)=u(;+h,t)-u&h,,>=,.

146 C. Y. CHAN and D. T. FUNG

It follows from the maximum principle for a weakly coupled parabolic system (cf. Protter and Weinberger [14, p. 1901) that v > 0 in (h, a/2) for each t E (0, T), and, thus, u is a strictly increasing function of x E (0, a/2).

The second part of the lemma follows from the symmetry of u.

The following result is a modification of theorem 4 of Chan and Kaper [6] from a scalar equation to a vector equation under different boundary conditions.

THEOREM 3. If lim,,, u&z/2, t) = q2, then lim,,,- ~,((a + 77)/2, t; a + a) = q2 for some finite time T.

Proof. Suppose

a+q f.42 ~ t;a+q

2 ’ > <q2 for all t > 0.

Then by lemma 2,

u,(x, t; a + rl) < q2

in (0, a + II) x (0, 00). Since fi(O, z) is a nondecreasing function of z and fi(O, z)

z + q2, we may choose positive numbers 6, to and E such that

fi(O,Z) 2 7 + ; + q*lcl for z L q2 - 6,

u2 ( >

;,t;a >q*-6 for t 2 t,,

2.4,(x, t; CI + q) 5 q2 - & for t 5 t, + 16qz 72

where C satisfies

f2(0, u,(x, t; a + rl)) - fi(0, 4(x, t; a + rl)) 2 Cu,(x, t; a + ?I)

for u,(x, t; a + q) 5 q2 - E. Let

It follows from (2.2), theorem 1 (c) and 1 (f), and lemma 2 that

24,(x, t; a + r/) 2 q2 - 6

on the parabolic boundary 8A. Since f2 is nondecreasing, we have

Huz(x, t; Q + rl) 5 -fz(O, u,(x, t; CI + q)).

From (2.3) and (2.4),

(H + CM.& t; a + V) 5 -f1(0, Ux, t; 0 + rl)) in A.

m as

(2.1)

(2.2)

(2.3)

(2.4)


Let

z(x,t)=q,-6-t (*-;)(f+-x)(t_to). Then, z = q2 - 6 on the parabolic boundary ?)A of A. In A,

+c/q2-s+ (x-;)(yL)(t-to,]

2 -2(t - to> - $ + JCJ q2 - 6 + $t - to) [ 1

ICI% >

.

BY (2.11,

Since

(H + 02 2 -fi(O, 2) in A.

q2 - 6 = 2(x, t) 5 u,(x, t; Q + Y/) on aA,

it follows from the maximum principle that

U,(X, t; L2 + ?I) 2 z(x, t) in A.

Now,

=q2.

This implies that

rq2,

and we have a contradiction. Thus, the theorem is proved.

Let !A, = (0, a) x (0, 00). We remark that from theorem 1 (e) and 1 (f), there exists a critical length a* such that u exists in C&, if a < a *. It is determined by the supremum of all values a such that a steady-state solution U exists. Theorem 3 implies that u2 reaches q2 somewhere in a finite time if a > a*.

3. BLOW-UP OF THE TIME DERIVATIVE

Let

co_ = 0,; x (0, T), ( > a

Co, = ( > - , a 2

x (0, T).

148 C. Y. CHAN and D. T. FUNG

LEMMA 4. In w_, u,~ 2 0. In w,, u,* 5 0.

Proof. Let

w(x, t) = u(x + p, t + T) - u(x + p, t) - u(x, t + T) + u(x, t)

for (x, t) E [0, (a - p)/2] x [0, T - t), where p (<a) and T (CT) are some positive numbers. Let

Wij(X9 t, = 1 l afw, t, 0)) do

.O drj ’

where r(x, t, a) = au(x, t + T) + (1 - a)u(x, t) with (T E [0, 11. Since

~f,(r(X, t, 0)) = ~ a~‘r(~~,t’ a)) [Uj(X, t + T) - Uj(X, t)], j= 1 J

it follows from integrating with respect to cr from 0 to 1 that

fj(U(X, t + 5)) - fj(U(X, t)) = jP, v/ij(X, f)I”jCx9 t + T, - uj(x, t)l-

ffw; = - I[fi(U(X + P, t + T)) - f;(u(x + P, t))l - [fi(W, t + 9) - h(u(x, t))ll

= ,il iWijCx + Pt t)[uj(x + P, t + T, - uj(x + p, t)l - Wij(X, t)[UjCX, t + @ - uj(x* t)lI*

By rearranging terms, we obtain

HW; + i Wij(X + P, t)Wj = - f: [Wij(X + P, t) - Wij(X, t)][Uj(X* t + t) - Uj(X, t)].

j=l j= 1

By the mean value theorem,

WijCx + P* t, - WijCxY f,

X id&(x + P, t + r) - u,(x, t + T)] + (1 - @t&c@ + P, t) - u,(x, t)]) da,

where (ok(~), A,(a)) lies on the line segment joining r(x, t, a) and r(x + p, t, a). By lemma 2, U,(X + p, t + T) - uk(x, t + T) and uk (x + p, t) - u,(x, t) are positive for 0 < x < (a - p)/2. Since the second partial derivatives of fj are nonnegative, we have

2

HWi + C Wij(X + P, t)Wj I 0 in (0, (a - p)/2) x (0, T - 7). j= 1

Again from lemma 2, w(x, 0) > 0 for 0 < x < (a - p)/2. By theorem 1 (c),

w(0, t) = u(p, t + 5) - u(p, t) > 0.


Because of symmetry, ~((a - p)/2, t) = 0. Thus, by the maximum principle for a weakly coupled parabolic system, w > 0 in (0, (a - p)/2) x (0, T - t). Since

Q(X, t) = lim - ,

we have uXt 1 0 in o_. The proof for uXt I 0 in 0, is similar.

The next result shows that there is exactly one quenching point.

THEOREM 5. If the solution u quenches at some finite time T, then the only quenching point is at x = a/2.

Proof. Let

cY(f) = 242 ;, t ( >

- ~2(-%, t) for 0 < x, < i.

Since uXt 2 0 in w_, it follows that a’(t) 1 0. Hence, a(t) is a nondecreasing function for each x, E (0, a/2). On the other hand, a(t) > 0 by lemma 2. Therefore, there exist positive constants I, (CT) and E such that a(t) 2 E for t 1 to. Thus, x, cannot be a quenching point. The theorem then follows from the symmetry of the solution.

To prove the blow-up of the time derivative, we need the following result.

LEMMA 6. For any positive constant T such that t < a and r < T (cm), there constant column vector c such that ut 1 c in [r, a - 51 x [r, T).

is a positive

Proof. Let 8sZ, denote the parabolic boundary of Sz,. Let 6 be the solution of the linear problem,

a = -f(O) in Q2,, ~=OonaQ,.

Since f is a nondecreasing function of its arguments, it follows from the maximum principle for a weakly coupled parabolic system that 6 5 u on Sz. Also, a proof analogous to that of lemma 1 (c) of Chan and Chen [8] shows that & is an increasing function oft for 0 < x < a. For any positive constant h (CT), let

d(x, t) = u(x, t + h) - u(x, t) - &(x, t + h) + 5(x, t).

Then,

Hd = -[f(u(x, t + h)) - f(u(x, t))] I 0 for (x, t) E (0, a) x (0, T - h),

d(x, 0) = u(x, h) - 6(x, h) 2 0 for 0 5 x 5 a,

d(0, t) = 0 = d(a, t) forO<t< T-h.

Thus, d(x, t) I 0 in (0, a) x (0, T - h). Hence,

u,(x, t) 2 &(x9 t) in Q.

150 C. Y. CHAN and D. T. FUNC

Since H&(x, t) = 0 in &, and St > 0 on the parabolic boundary of (t, a - T) x (5, T + 7), we

have cr 2 c on [r, a - t] x [T, T + t] for some positive constant column vector c. Hence, the lemma is proved.

Our next result gives the blow-up of the time derivative of the solution.

THEOREM 7. Suppose u&/2, t) + q2 as t + T- for some finite time T, and there exists a function Pi(w) in CIO, qz] fl C’(O, q2) such that

Pl > 0, Pl(q2) = 0, ~[P,(~2)_I-l(% 3 M/h 2 0 for @,,ud E @,4J x (0, qdr (3.1)

p; s 0, and p;’ 5 0. Then, u&7/2, t) -+ co as t + T-. If in addition, u,(a/2, t) + q1 as t -+ T-, and there exists p2(w) in CIO, qJ 17 C’(O, qJ such that

Pz > 0, P2(41) = 0, ~[P2@l)“ml~ %)l/au, 1 0 for (u,, 4) E (0,4d x (0, d, (3.2)

pi I 0, andp; I 0, then u,(a/2, t) + 00 as t + T-.

Proof. Let

sr(x, t) = Pl@,(X, m&,(x, t).

By theorem 1 (c), and lemmas 2 and 4,

Hs, = PX~2)(~2,h, + P;(~2)~2,%t + 2P;(W,Xu,Xt - P;@2)~2,%, - [Pl(~2).f&Q > %)lf

5 PX~*)(~2,~1, - &,,U2,) - [P1042)“fl(~l 9 U2)lr.

Since a[p,(u,)fi(ui, u,)]/du, 1 0, we have

Hs, 5 P;(~2)(~2*x~l, - h,U2,).

Thus,

It follows from theorem 1 (c) and pi I 0 that

Hs 1

+ P;@2).ml~ u2) s I ()

P&42) 1 * (3.3)

From lemma 6, there is a positive constant column vector c such that u, L c in [r, a - T] x [T, T) for some positive constant r. For any positive constant E satisfying E < a/2 - t and E < T - T, the lines x = a/2 - E and x = a/2 + E have no quenching points. Thus, there exists a positive constant c3 such that c3 5 p1(u2) < 00 on the parabolic boundary X2, of a,, where

Thus,

s1 = P1@2)%, 2 C3Cl on an,.


By the weak maximum principle, s1 L c3c1 (>0) in Q. Since

= Pl(q2) = 0,

we have ull(a/2, t) -+ co as t + T-. To prove the second part of the theorem, let s2(x, t) = pz(u,(x, t))uz,(x, t). Similar to (3.3),

we obtain

Also, there exists a positive constant c, such that c, I p2(u1) < CO on the parabolic boundary aa,. Thus,

s2 = P2@&2, 1 c4c2 on asz,.

Hence, s2 L c, c2 (>O) in Q2,. Since

= P2GA) = 0,

it follows that u2t(a/2, t) + CD as t -, T-. Hence, the theorem is proved.

We remark that (3.1) implies lim,,,Ji(u) = 00, and (3.2) implies lim,,,,J2(u) = co. We also note that the proof of theorem 7 does not depend on u2 2 u1 and q1 2 q2. Since condition (iii) is used to show u2 2 u1 in SJ, it is not needed in proving the blow-up of the time derivative.

For illustration of theorem 7, let us consider the following example. In problem (1.6), let

J-1(%, u2) = (1 - w1 + hl Ul (3.4)

for some nonnegative number h, . We may take

Pl(U2) = 1 - 4 with n 2 2(1 + h,).

Then,

$ [Pl(~Z)_fd u1, u2)] = 1 + 22.4, + 3u; + . . . + (n - 1)q2 - nh, U1 u;-’ 2

n > - ulu;-‘(n - 1 - 2h,) > 0

2

since 0 < u < 1. If u2(a/2, t) + 1 as t + T- for some finite time T, then u,,(a/2, t) -+ 00 as t --* T- by theorem 7. To illustrate the second part of the theorem, let

f2@1, u2) = (1 - w1 + h,u, (3.5)

for some nonnegative number h2 with h, < h2. Here, we may take

Pz(U,) = 1 - u;” with m 2 2(1 + h,).

If u,(a/2, t) + 1 as t -+ T-, then u,(a/2, t) --t 00 as t --t T-. We note that the forcing term f(u) given by (3.4) and (3.5) was considered by Chan and

Chen [8] in illustrating the computation of the critical length for global existence of the solution.

152 C. Y. CHAN and D. T. FUNC

REFERENCES

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

2. 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).

3. CHAN C. Y. & KWONG M. K., Existence results of steady-states of semilinear reaction-diffusion equations and their applications, J. diff. Eqns 77, 304-321 (1989).

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

5. CHAN C. Y. & CHEN C. S., A numerical method for semilinear singular parabolic quenching problems, Q. appf. Math. 47, 45-57 (1989).

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

7. CHAN C. Y. & COBB S. S., Critical lengths for semilinear singular parabolic mixed boundary-value problems, Q. uppl. Math. 49, 497-506 (1991).

8. CHAN C. Y. & CHEN C. S., Critical lengths for global existence of solutions for coupled semilinear singular parabolic problems, Q. uppl. Math. 47, 661-671 (1989).

9. KAWARADA H., On solutions of initial-boundary problem for u, = U, + l/(1 - u), Publ. Res. Inst. Math. Sci. 10, 729-736 (1975).

10. CHAN C. Y. & KWONG M. K., Quenching phenomena for singular nonlinear parabolic equations, Nonlinear Analysis 12, 1377-1383 (1988).

11. ACKER A. & KAWOHL B., Remarks on quenching, Nonlinear Analysis 13, 53-61 (1989). 12. FRIEDMAN A. & MCLEOD B., Blow-up of positive solutions of semilinear heat equations, Indiana Univ. math. J.

34, 425-447 (1985). 13. DENG K. & LEVINE H. A., On the blowup of U, at quenching, Proc. Am. math. Sot. 106, 1049-1056 (1989). 14. PROTTER M. H. & WEINBERGER H. F., Maximum Principles in DifferentialEquations. Springer, New York (1984).

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