solution profiles beyond quenching for degenerate reaction-diffusion problems

Pergamon Nonlinear Analysis, Theory, Methods & Applications, Vol. 24, No. 12, pp. 1755-1763, 1995

Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved

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S O L U T I O N P R O F I L E S B E Y O N D Q U E N C H I N G F O R

D E G E N E R A T E R E A C T I O N - D I F F U S I O N P R O B L E M S

C. Y. CHAN and P. C. KONGt

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

(Received 1 November 1993; received for publication 18 August 1994)

Key words and phrases: Beyond quenching, degenerate reaction-diffusion problem, steady-state solution.

1. I N T R O D U C T I O N

Let T < o0, 1~ =--- (0, a) x (0, T), 0II be the parabolic boundary ([0, a] x {0}) U ({0, a} X (0, ~)) of II, and L u = Uxx--xqut , where q is any real number. Floater [1] studied the degenerate equation L u = - u p in II subject to certain nonnegative initial data and zero boundary conditions. He proved that for 1 < p < q + 1, the solution blows up at x = 0 in finite time. This contrasts with a result of Friedman and McLeod [2], who showed that when q = 0, the blow-up set is a compact subset of (0, a), and hence, it is bounded away from x = 0 and from x = a. Recently, Chan and Kong [3] studied the problem,

L u = - f ( u ) in l l , u = 0 on c~II (1.1)

for q ~ 0, where f e C 2 ([0, c)) for some positive constant c such that f(0) > 0, f" > 0, f " > 0, l i m u _ ~ c - f ( u ) = ~ , and f ~ f ( u ) d u = M for some positive constant M. They showed that u quenches somewhere in [c2 / (2Ma) , a / 2 ] for q > 0, and somewhere in [ a / 2 , a - c2 / (2Ma)] for q < 0. That is, the set of quenching points lies in a compact subset of (0, a). They also show that u t blows up at each quenching point, and the critical length a* is the same as that when q = 0 .

Our main purpose here is to study what happens beyond quenching of the problem (1.1). We consider the following degenerate reaction-diffusion problem

L u = - f (u ) x~u<c} in II, u = 0 o n 0 l l , (1.2)

where q > - 1, q ~ 0, and

(10 i f u ~ S , Xs = if u q~ S,

is the characteristic function of the set S. In addition to the above assumptions on f , we assume that for some positive constants K~(> 1), K 2 and K3, and some constant y in (0, 2),

, C - - U 2

_F I } f ( x ) ds < min K2(c - u ) f ( u ) , K3(c - u) ~ . (1.4) ~U

tCurrent address:: Carson-Newman College, Jefferson City, TN 37760, U.S.A.

1755

1756 c.Y. CHAN and P. C. KONG

A function u is a weak solution of the problem (1.2) if and only if: (a) u ~ C([0, to]; LI((0, a)) )n L°~((0, a) x (0, to)) for each t o > 0; (b) for any g ~ C E J ( ~ ) such that g has a compact support with respect to

g(O, t) = 0 =g(a , t), t and

0f f0 a Ff af( uL*g dx dt + u)xtu < c}g dx dt = O, • "0 " 0

where L* = c~2/c~x 2 +xqo~/o~t is the adjoint operator of L. We prove that the problem (1.2) has a weak solution such that

(1.5)

u(x,t) E C 2 ' l ( { u < c } n ~']) 0 cl'°(~-~)n C ( ~ ) , u _<c in ~ , (1.6)

u is nondecreasing with respect to t in {u < c} n 12. (1.7)

This extends a one-dimensional result of Phillips [4] to the problem (1.2). He studied the problem, which (in our notation) is given by

A/a, --/~, = --A(1 - jLt,)-/3X{/~ < 1} in D X [O,oo),

/z =/.%(x) < 1 on D X {0}, /z = 0 on 3D × (0, ~),

where A and/3 are positive constants with /3 < 1, D is a bounded n-dimensional domain, D is its closure, c~D is its boundary, and /z 0 and aD are of class C 2+ ~ with /z 0 = 0 on ¢D. We remark that the function (1 - u ) -# satisfies the assumptions (1.3) and (1.4) with c = 1. He established existence of a weak solution ~ with continuous/Xx, and posed its uniqueness as an open question. To avoid this question of uniqueness, Fila et al. [5] considered only the (unique) minimal weak solution /x. They showed that every element of the ~ l imi t set of the minimal weak solution (among solutions obtained via regularization) is a weak steady-state solution of the problem. If D is a ball and A is large enough, then the minimal weak solutions (corresponding to different /x 0) converge (in the sup-norm) to the unique weak steady-state solution with a set {/z - 1} of positive measure. By using a completely different approach, we show here that as t tends to infinity, all weak solutions (not necessarily obtained via regularization) of the problem (1.2), having properties (1.6) and (1.7), tend to a unique steady-state (nonclassical) solution, which is independent of q. This extends the results of Chan and Ke [6], who considered the case q = 0. In the special case when f ( u ) = (1 - u) -~, Chan and Ke proved the conjecture of Levine [7] for a > a*. We remark that the method of Chan and Ke applies to phenomena with a single-point quenching while ours does not require this property.

2. EXISTENCE OF WEAK SOLUTIONS

For any constant e ~ (0, 1), let

¢ - - U

L ( U ) =f(u) ~ f ( u ) + C -- U"

Solution profiles beyond quenching for degenerate reaction-diffusion problems 1757

Then, lim, ~ 0f,(u) =f(u)xtu < ~, and f , (u ) <f (u ) for all ~ > 0. From (1.3), f ' ( u ) < (K 1 - ~) /e2, and hence for any E, f" is bounded above. Let

[-c-~'u for q > 0, L 'u = ~ Lu for - 1 < q < 0,

where S a" = a2/Ox 2 - (x + E) q a /a t . We consider the following regularized problem,

L'u" = - f , ( u ' ) in 12, u" = 0 on c~12. (2.1)

Since 0 is a lower solution and c is an upper solution of the problem (2.1), the following existence result follows from theorem 4.2.2 of Ladde et al. [8, p. 143].

LEMMA 1. The problem (2.1) has a solution u ' ~ C2÷~'1+~/2(~).

The proof ol the following result is similar to that of lemma 1 of Chan and Kaper [9] since there is a corresponding strong maximum principle for the operator L (cf. Friedman [10, p. 39]).

LEMMA 2. The problem (2.1) has at most one solution. In 12, 0 < u" < c and u 7 > 0.

We note that it follows from lemma 2 and the strong maximum principle that u 7 > 0 in 12. We now show that u" is a strictly decreasing function of ~.

LEMMA 3. If 0 < ~1 < ~2 < 1, then u ~' > u ~2 in IL

Proof. From E 1 < E2, we have f,,(u) >f,2(u). Let w(x, t) = u' l(x, t) - u'2(x, t). For q > 0, it follows from u~ > 0 and (x + E2)q > (x + E1)q in f~ that

( S a'~ +f',~( ~ ))w < 0 in 12, w = 0 on a12,

where ~ lies between u'~(x, t) and u ' f fx , t). By the strong maximum principle, w > 0 in 1"~. For q < 0, we have

( L + f ' ( r l ) ) w < 0 in f~, w = 0 on 01~,

where 7/lies between u'l(x, t) and u':(x, t). Thus, u'~(x, t) > u'2(x, t) in fL Since 0 < u" < c in 12, and u ' ~ c 2 ' t ( ~ ) is strictly increasing as ~ decreases, it follows from

the Dini theorem that u" converges uniformly on f~. This limit, denoted by u(x, t ) , is continuous on ~ .

With lemmas 1, 2 and 3, an argument similar to that in the proof of theorem 1 of Phillips [4] gives the following result.

THEOREM 4. The limit u(x, t) satisfies Lu = - f ( u ) in {u < c} f') 12 in the classical sense. From lemma 2, 0 < u(x, t) < c in IL By theorem 4, u(x, t) ~ C2'1({u < c} tq 12). Since u t > 0

in 12, it follows that u,(x, t) > 0 in {u < c} f3 12. To prove existence of a weak solution of the problem (1.2), we need the following two lemmas.

1758 C.Y. CHAN and P. C. KONG

LEMMA 5. In {u < c}O ~ , (Ux(X, t)) 2 < 2]'C(x,of(s)ds.

Proof. Let us prove the lemma for the case q > 0 since its proof for q < 0 is similar. We show first that

f ( u ~ ( x , t ) ) z < 2 f~(s)ds for t > O . (2.2) • "u (x.t)

For each fixed t > 0, it follows from u ' (0 , t) = 0 = u'(a, t) and u" > 0 in ~ that u~(0, t) > 0 and u~(a, t) < O. For any x ~ [0, a]:

(a) if u~(x, t) = 0, then (2.2) follows; (b) if u~(x , t )> 0, then by the continuity of u~, there exists some x I > x such that

u~(xl, t) = 0 and u~(.,t) > 0 in (x, xl). From (2.1)

fx, Yx" U~x(r , t )u~(r , t )dr= (r + e)quT(r , t )u '~(r , t )dr - f , ( u ' ( r , t ) ) u ~ ( r , t ) d r . "X

Since u t > 0 on ~ , we have

fu'(xl,t) [ j ( u ~ ( x ' t ) ) 2 < 2 1~ f , ( s ) d s _< 2 f , ( s ) ds;

" u (x,t) " u (x,t)

(c) if u~(x , t )< 0, then by the continuity of u~, there exists some x 2 < x such that u~(x2, t) = 0 and Ux(, t) < 0 in (x2, x). Then

I~U "(X,t) t~U "(X2,t)

(U'x(X,t))2<_ - 2 ] , f , ( s ) d s = 2 Ju'(x,t) f , ( s ) d s , • " u ( X z , t )

f rom which (2.2) follows. Since u~' converges to u x in {u < c} f'l lq, the lemma is proved.

Let

.~ E * 032 69 = + ( X "Jr- E) q

a X 2 - ' ~ '

L~ , _- [ . ~ ~* for q > O, ~L* f o r - l < q < O .

They are, respectively, the adjoint operators of 2 ~ and LL

LEMMA 6. For some constant K 4 independent of

f ( u ( x , t ) )xi u < c)g(x, t) dx dt < K 4 < ~. (2.3)

Proof. We note that

g L "u" =

" q ~ u ' . ~ ' * g (Uxg) x - (U'gx) x - ( (x + ~) u g)t + for q > 0,

(u~g) x - (U'gx) x - (xqu'g)t + u'L*g for - 1 < q < 0.


Since g(x, t) vanishes at x = 0 and x = a, and has a compact support with respect to t, it follows f rom (2.1) that

f~(u ' )gdxdt = - u 'L '*gdxdt . JO " 0

Without loss o f generality, g may be taken to be nonnegat ive since g is bounded below. F r o m 0 < u" < c in ~ , 0 < (x + ~)q < (a + 1) q for q > 0, and f~ x q dx < ~ if - 1 < q < 0, we have

fo fo~f~(u ' )gdxdt<K4(g ,c ,a ,q)<~

for some constant K 4 independent o f ~. Since l im, ~ 0f , (u) =f(u)xt, < c}, (2.3) follows by using the Fa tou lemma.

We modify the idea of Phillips [4] to prove the following result.

THEOREM 7. "lqae limit u(x, t) satisfies (1.5).

Proof. Let us pick a funct ion ~(s) ~ C~(0, ~) such that

~ ( s ) = { s - 1 for s > 2, 0 for s < ½,

~ ' > 0 and ~" > 0. For any constant h > 0, let

Since u ~ ~ u uniformly as ~ ~ 0, and ~h(S) ~ SXl, > 0/ as h ~ 0, we have

~o foa~Oh(C-u~)L'*gdxdt

~Oh(C -- u)L*g dx dt as E -~ 0

--* (c-u)xtc_,>o}L*gdxdt as h ~ 0

= (c - u)L*g dx dt

fo foYo = c(gx(a,t) -gx (O, t ) )d t - uL*gdxdt.

1760 C.Y. CHAN and P. C. KONG

On the other hand, it follows from the properties of g and u '(0, t ) = 0 = u ' ( a , t) that

0;f0 0; ~h(c - u ' ) L ' * g dx dt = ~oh(c)(gx(a, t) - gx(O, t)) dt

+ ~ ' : ~ ( c . . . . . 2 - u )tu x) g d x d t

+ ~o'h(c - u~)f~(u~)g dx dt • "0 "0

~ ~oq~h(c ) (gx (a , t ) -g~(O, t ) )d t

foe + ~'~(C - - U ) X { ( h / 2 ) < _ c _ u < 2 h } U 2 g d x d t

gE + q~'h(c-u)X{(h/2)<_c_u}f(u)gdxdt as E--* 0.

As h --* 0, q~h(c) --* c. Hence as h ---> 0, the above first term tends to

Let K 5 denote m a x , " . Then

By lemma 5 and (1.4),

~o C(gx(a, t) - gx(O, t)) dt.

K5 ~O'h ( S ) <~ ' - h X{( h /2) 4 s < 2h}"

~o foa~Onh(¢ -- U)X{(h/2)< - 2h}U2g d x dt ¢ - u <

< h • "O ~0

<- 4KzKs u)x{(h/2)< c-u < 2h}g dx dt, "0 "~0

which, by lemma 6, tends to 0 as h ~ 0 . Since Iq~; , (c-u) l< 1, and ~'h(c-u)--*X{c-u>O} as h ~ 0, it follows that as h ---, 0,

q~'h(C--U h / Z ) ~ c _ , } f ( u ) g d x d t ~ u u<c}gdxdt , • * o ,t o

which, by lemma 6, is bounded. Hence, the theorem is proved.

LEMMA 8. In fL ux is continuous.

Proof. For each fixed t > 0, it follows from (2.2), f , (u ) <f(u ) and (1.4) that

lux'l-< v /2g3(c - u ' ) ~' .


By lemma 2, (c - u ' ) > 0 in II. Hence

I ( c - <_ 2v/-UU3 .

For any x and y such that 0 < x < y < a , we have

fx If' fx ¢ u ' ) - Y / 2 u : d S < I(c u ' ) - Y / 2 u : l d s < ~ Y

which gives

I(c - u ' ( y , / ) ) t 2 - ~)/2 _ (c - u ' ( x , t )) t2- v)/2[ < K61y _ xl,

where g 6 = V/-2K3 (2 - 3,)/2. Since u" converges monotonically to u on ~ , it follows from the Dini theorem that (c - u ' ( x , t)) ~2- v)/2 converges uniformly to (c - u ( x , t)) t2- 7)/2. Thus

[(c - u(y , t ) ) (2- 7)/2 _ _ ( C - - U(X, t ) ) (2- 7)/21 _ g61y - x]. (2.4)

If u(x o, t o) = c at some point (x 0, t o) ~ l-l, then it follows from (2.4) that

0 <_ C -- U(X, t o) <_ K 2/(2- 7) Ix - Xol 2/(2- v).

Hence

lu(x't°)-u(x°'t°)lx=-~o < K26/t2-7)lX-Xo, v/t2-')"

This implies that Ux(X o, t o) exists and equals to 0. It follows from thoerem 4 that u x is continuous in l-l.

From the above discussion, we have shown that the problem (1.2) has at least a weak solution having properties (1.6) and (1.7). We note that for any weak solution u (not necessarily obtained via regularization) having the property (1.7), we have u > 0 in ~ . Also, it follows from the strong maximum principle that for any weak solution u satisfying the partial differential equation in the problem (1.2), we have u < c in 1).

3. BEYOND Q U E N C H I N G

The first finite time ~- when quenching occurs is called the quenching time. For t >_ ~-, let

b( t ) = inf{x: u(x , t) = c},

B ( t ) = sup{x: u(x , t) = c},

b* = lim b( t ) , t--*~

B* = lim B( t ) . t - ~ 0 0

We would like to show that as t tends to infinity, all weak solutions of the problem (1.2) having properties (1.6) and (1.7) tend to the unique solution of the steady-state problem

W ( x ) = c for b* < x < B~*, (3.1)

W" ( x ) = - f ( W ( x ) ) for 0 < x < b~, W(0) = 0, W(b* ) = c, (3.2)

W " ( x ) = - f ( W ( x ) ) for B* < x <a, W ( B * ) =c , W(a) = 0, (3.3)

1762 c.Y. CHAN and P. C. KONG

where

l foe b* = v~ (g(c) - g ( s ) ) 1/2 ds,

g(s) = fo~f(~b)d~b,

B* = a - b*.

Henceforth, we use u to denote any weak solution having properties (1.6) and (1.7). A proof similar to those of lemmas 2 and 6 of Chan and Ke [6] gives the following result.

LEMMA 9. The function b(t) is nonincreasing while the function B(t) is nondecreasing; furthermore, b* > b* > 0, and B* < B* < a.

Since u ( < c) is continuous and nondecreasing with respect to t, it follows from the Dini theorem that u(x, t) converges uniformly to a limit, limt_~= u(x, t), which is continuous on [0, a]. Let

U(x) = lim u(x, t). t - - ~ oc

A proof similar to that of lemma 3 of Chan and Ke [6] gives the following result.

LEMMA 10. For x ~ (0, b*), u(x, t) converges uniformly to the solution of (3.2) with b* = b*. For x ~ (B*, a), u(x, t) converges uniformly to the solution of (3.3) with B* = B*.

Since we may not have a single-point quenching, and have not proved that the parabolic boundary of {(x, t) ~ ~ : u(x, t) = c} is continuous, we cannot use the argument of Chan and Ke [6] to deduce the following result from lemma 10.

LEMMA 11. For x~[b*,B*] , U ( x ) - c .

Proof. Let us suppose that there exists some x 0 ~ (b*, B*) such that U(x o) < c. By the continuity of U, there exists an interval (Xl, x2)with b* < x 1 < x 0 < x 2 < B* such that U(x 1) = c = U(x 2) and U(x) < c for x ~ (Xl, x2). Since u t > 0 for u < c, we have u(x, t) < c in {(x, t): x ~ (x 1, x 2) and 0 < t}. This implies that

Let

where

Lu = - f ( u ) for x 1 "(X <X2, 0 < t.

f x

F(x , t ) = ~qG(x; i~ )u( ~ , t )d~, x I

[ ( X z - X ) ( ¢ - x l )

G(x;~ ' )= (x 2 - ~ ' ) ( x - x 1)

X 2 - - X 1

for x 1 _< ~'_<x,

for x _< ~'_~<x2,

(3.4)


is Green's function corresponding to the operator d2 /dx 2 subject to the first boundary conditions. Since u(x, t) is nondecreasing with respect to t, it follows from (3.4) that lim t ~ Ft(x, t) > O. By a direct calculation,

X -- X 1 fx2 X 2 - -X U ( X l , t ) + U(x2,t) + G(x; ~)f(u(~,t))d~. Ft(x , t ) = - u ( x , t ) + x 2 - x 1 X2 --X1 "~xl

Since f is nondecreasing, it follows from the monotone convergence theorem, the continuity of f , and U(x 1) = c = U(x 2) that

fx2 l imFt (x , t ) = - g ( x ) + c + G(x; ~ ' ) f ( g ( ~ ) ) d ~ . (3.5) X 1

To show that lim t_.~ Ft(x, t ) = O, let us suppose that the limit is (strictly) positive at some point x ~ (xl , x2). Then as t tends to infinity, F(x, t) increases without bound there. This implies that u reaches c at some finite time, and contradicts U(x) < c for x ~ (x 1, x2). Thus lira t _. ® Ft(x, t) = 0. From (3.5),

U(x)=c+ G(x;~)f(g(~))d~>c forx~(xt,x2).

This contradiction proves the lemma.

From lemmas 10 and 11, we obtain the following result.

THEOREM 12. As t tends to infinity, all weak solutions of the problem (1.2) having properties (1.6) and (1.7) tend to the unique steady-state profile given by (3.1), (3.2) and (3.3).

R E F E R E N C E S

1. FLOATER M. S., Blow-up at the boundary for degenerate semilinear parabolic equations, Archs. ration Mech. Analysis 114, 57-77 (1991).

2. FRIEDMAN A. & MCLEOD B., Blow-up of positive solutions of semilinear heat equations, Indiana Univ. math. J. 34, 425-447 (1985).

3. CHAN C. Y. & KONG P. C., Quenching for degenerate semilinear parabolic equations, Appl. Analysis 54, 17-25 (1994).

4. PHILLIPS D., Existence of solutions of quenching problems, Appl. Analysis 24, 253-264 (1987). 5. FILA M., LEVINE H. A & VAZQUEZ J. L., Stabilization of solutions of weakly singular quenching problems,

Proc. Am. math. Soc. 119, 555-559 (1993). 6. CHAN C. Y. & KE L., Beyond quenching for singular reaction-diffusion problems, Math meth. Appl. Sci. 17, 1-9

(1994). 7. LEVINE H. A., Quenching nonquenching, and beyond quenching for solution of some parabolic equations,

Annali. Mat. pura appl. 155(4), 243-260 (1989). 8. LADDE G. S., LAKSHMIKANTHAM V. & VATSALA A. S., Monotone Iterative Techniques for Nonlinear

Differential Equations, Pitman Advanced Publishing Program, Boston (1985). 9. CHAN C. Y. & KAPER H. G., Quenching for semilinear singular parabolic problems, SlAM J. math. Analysis 20,

558-566 (1989). 10. FRIEDMAN A., PartialDifferentialEquations of Parabolic Type. Prentice-Hail, Englewood Cliffs (1964).

solution profiles beyond quenching for degenerate reaction-diffusion problems

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