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Necessary and Sufficient Conditions for the Unique Solvability of a Nonlinear Reaction-Diffusion Model
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
228, 483]494 Ž1998.
AY986165
Necessary and Sufficient Conditions for the Unique Solvability of a Nonlinear Reaction-Diffusion Model Jeffrey R. Anderson Department of Mathematics and Statistics, Winona State Uni¨ ersity, Winona, Minnesota 55987 Submitted by John La¨ ery Received December 30, 1997
It has been known for some time that a nonlinear reaction-diffusion model, with Dirichlet boundary conditions, is uniquely solvable if the reaction term satisfies an appropriate Lipschitz condition. However, as recently shown for an absorption model, such a condition is not necessary. We establish a uniqueness result which, in the case of reaction and diffusion governed by power laws, is in fact both necessary and sufficient for the unique solvability of the model. The improvement that is needed on the above-mentioned Lipschitz condition occurs in the so-called fast diffusion model. Q 1998 Academic Press
1. INTRODUCTION The goal of this investigation is to obtain conditions that guarantee unique solvability of the nonlinear reaction-diffusion model, u t s D f Ž x, t , u . q f Ž x, t , u . usc u s u0
on Ž V . T
on V T
Ž RD.
on V = 0 4 ,
and to study the extent to which these are necessary. Here, V ; R N is a bounded domain with C 2 boundary, V T ' V = Ž0, T ., and c G 0, u 0 G 0, with u 0 g L`Ž V ., c g L`ŽŽ0, T ... We shall assume f Ž?, ? , 0. s 0 and, hence, consider only nonnegative solutions of the model. The parabolicity condition, fu ) 0 for u ) 0, and smoothness conditions required of f and f 483 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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JEFFREY R. ANDERSON
are exactly those that are sufficient for the construction of a local weak solution. In the case of power laws, f s u m , f s au p , these allow all positive powers m, p ) 0. If f is assumed to be Žlocally. Lipschitz continuous in u, then the uniqueness of solutions of ŽRD. has appeared in many articles, e.g., w1]4, 7x. In the case of an ‘‘absorption law,’’ i.e., a F 0, it has been shown that Lipschitz continuity is no longer necessary w8x. Moreover, as will be established, the Lipschitz condition on f is not necessary, even in the reactive case, a ) 0. Toward developing a uniqueness result for ŽRD. that is both necessary and sufficient in the instance of power laws, we propose the following modified Lipschitz condition on f : There exist positive constants CM , DM such that f Ž x, t , u . y f Ž x, t , ¨ . F CM f Ž x, t , u . y f Ž x, t , ¨ . q DM w u y ¨ x
Ž ML .
for every 0 F ¨ F u F M. We show that ŽML. is sufficient for the uniqueness of all nonnegative solutions of ŽRD. provided CM F l 0 for some M ) 0, and l 0 is the smallest Dirichlet eigenvalue for V. Such a condition is satisfied by the power laws if a F 0 or p G min m, 14 Žand a F l0 in the situation p s m - 1.. The remaining cases of a, m, p are contained in two nonuniqueness results for ŽRD., in which we prove that the trivial solution is not unique. Thus, ŽML. with CM F l 0 is both necessary and sufficient for the unique solvability of the power law version of ŽRD.. This is our main result. It can be seen that ŽML. differs significantly from the standard Lipschitz condition in a model of ‘‘fast’’ diffusion, e.g., 0 - m - 1. In the next section we give a precise definition of a weak Žlocal. solution of ŽRD. and state our main results on uniqueness and nonuniqueness for such solutions. A maximal weak solution is then constructed via monotone limits of solutions to suitably ‘‘regularized’’ models. Although the construction of such a solution is not new Že.g., see w1, 2, 6x., it is key to the development of uniqueness and nonuniqueness for ŽRD.. Having constructed the maximal solution, U ' UŽ x, t; c , u 0 ., we have u F U for any other solution, u s uŽ x, t; c , u 0 .. In Section 3, we prove the unique solvability of ŽRD. by establishing that the reverse inequality is also true. A benefit of the monotone construction of U is that once solutions of ŽRD. are known to be unique, solution and subsolution comparison principles follow immediately. As our method makes use of the key fact U G u, this work does not appear to allow development of a general supersolution comparison result. In Section 4, we utilize the construction of U and show that, under suitable conditions, a subsolution argument can be fashioned that yields
NONLINEAR REACTION-DIFFUSION MODEL
485
UŽ?, ? ; 0, 0. k 0. Thus, the trivial solution is not unique. Whether or not solutions are unique if u 0 k 0, c ' 0 is yet an open problem. On the other hand, we give instances in which ŽRD. is not uniquely solvable for trivial initial and Dirichlet data, has a nontrivial equilibrium state, w, and, yet, uŽ?, ? ; 0, w . s w for any solution, u, of ŽRD.. That is, we have the uniqueness of solutions if c ' 0 and u 0 ' w. In the final section, we present a possible generalization of this work to a reaction-diffusion model with convection. However, in the presence of convection, a complete study of necessary and sufficient conditions for the unique solvability of the model is left for a future study.
2. MAXIMAL SOLUTION OF ŽRD. AND STATEMENT OF THE MAIN RESULTS To construct a maximal solution of ŽRD., assume Ži. V T = R. Žii. Žiii. Živ.
f, f , D x f , and the components of =x f are all continuous on The partial derivatives fu , f u are continuous on V T = R R 04 . fu ) 0 for u ) 0. f Ž x, t, 0. s f Ž x, t, 0. s 0.
Here, D x and =x are used to denote the Laplacian and gradient operators, respectively, each taken with respect to the x variables only. ŽContinuity in x and t can actually be replaced with a weaker condition, such as inclusion in L` , and the construction used herein is still valid w1x.. We also assume that V is a bounded domain in R N with C 2 boundary, i.e., V is locally representable by twice continuously differentiable functions. The outward pointing unit normal to V is denoted by n. As the differential equation is potentially of degenerate parabolic type, we do not expect classical solutions of ŽRD. in general. The notion of a weak Žlocal. solution adopted here is the same as in previous work w1x. To state the definition, let the class of nonnegative test functions be defined as T ' j : j t , D j , < =j < g L2 Ž V T . , j G 0, and j < Ž V . T s 0 4 . DEFINITION 2.1. A function uŽ x, t . is called a subsolution Žsupersolution. of ŽRD. on V T if the following all hold: Ži. Žii.
u g L`Ž V T .. u F ŽG. u 0 on V = 04 .
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JEFFREY R. ANDERSON
Žiii. u F ŽG. c on Ž V .T . Živ. For every t g w0, T x and every j g T ,
HV u Ž x, t . j Ž x, t . dx F Ž G. HV u Ž x . j Ž x, 0. dx 0
t
H0 H V f Ž x, s, c Ž s . . =j ? n dS
y
t
H0 HV u j
q
s
x
ds
q f Ž x, s, u . D j q f Ž x, s, u . j 4 dx ds.
A function uŽ x, t . is called a Žlocal. solution of ŽRD. if it is both a subsolution and a supersolution of ŽRD. on V T for some T ) 0. To construct such a solution, define m Ž k . ' min
½
min
V T= w0, 2 k x
wDxf q f x, 0
5
and let u k denote the solution of u t s D f Ž x, t , u . q f Ž x, t , u . y m Ž k . uscqk u s u0 q k
on V T
on Ž V . T
Ž RDk .
on V = 0 4 .
T ) 0 may be chosen sufficiently small in such a way that there exists a solution of ŽRDk . on V T for every 0 - k F 1, and 5 u k 5 ` is bounded independently of k. Furthermore, k F u k F u l for 0 - k F l, and a subsolutionrsupersolution comparison theory holds for ŽRDk . w1x. As u k is monotone in k, we may define U ' lim k ª 0q u k , and it is easy to see that U is a solution of ŽRD.. Furthermore, if u is a solution of ŽRD. such that u s U on Ž V .T and on V = 04 , then
HV Ž u y u s
k
. j Ž x, t . dx
t
H0 HV Ž u y u
k
. j s q f Ž x, s, u . y f Ž x, s, u k . D j 4 dx ds
q
t
f Ž x, s, u . y f Ž x, s, u k . j q m Ž k . j 4 dx ds
y
t
f Ž x, s, c . y f Ž x, s, c q k . =j ? n dS x ds
H0 HV H0 H V
HV j Ž x, 0. dx.
yk
NONLINEAR REACTION-DIFFUSION MODEL
487
With F k and Fk defined so that
Ž u y u k . F k s f Ž x, s, u . y f Ž x, s, u k . and
Ž u y u k . Fk s f Ž x, s, u . y f Ž x, s, u k . , we have
HV Ž u y u
t
k
. j Ž x, t . dx F H
0
HV Ž u y u
k
. j s q F k D j q Fk j 4 dx ds.
Note here the use of yD j ? n G 0 on Ž V .T and ymŽ k . G 0 to derive this inequality. The choice of j according to j G 0, j s 0 on Ž V .T , and j s q F k D j q Fk j s 0 now yields u F u k . ŽIf necessary, F k and Fk may be smoothed out appropriately to obtain j , and passage to the limit as the smoothings are removed may be justified w1, 2x.. If u is a subsolution of ŽRD., the above argument shows that u F u k is generally true if the inequality holds on V = 04 j Ž V .T . Thus, U is the maximal solution of ŽRD., and this solution satisfies a subsolution comparison theory. To establish uniqueness for solutions of ŽRD., it only remains to prove that the reverse inequality is also true. Our result in this direction is the following. THEOREM 2.1. If Ž ML. is satisfied and CM F l 0 for some M ) 0, where l0 ) 0 is the smallest Dirichlet eigen¨ alue for V, then the Ž nonnegati¨ e . solution of Ž RD . is unique. Furthermore, if c 1 G c 2 G 0, u 0 G ¨ 0 G 0, u s uŽ x, t; c 1 , u 0 . is a solution of Ž RD ., and ¨ s ¨ Ž x, t; c 2 , ¨ 0 . is a subsolution of Ž RD ., then u G ¨ . Toward an investigation of multiple solutions for ŽRD., we introduce the following rather elaborate condition, which may be applied to the power laws. ŽHere the label ŽNU. is used to denote a sufficient condition for ‘‘nonuniqueness.’’.
Ž i.
There exists FM Ž t , k . such that f Ž x, t , k¨ . G ¨ FM Ž t , k .
Ž ii .
for x g V , t g Ž 0, T . , 0 - ¨ F M, k G 0. There exists g Ž t . such that g Ž 0 . s 0 with 0 - g 9 Ž t . F FM Ž t , g Ž t . . for t g Ž 0, T . .
Ž iii .
There exists h Ž t . such that h Ž 0 . s 0, h9 Ž t . ) 0 for t g Ž 0, T . ,
Ž NU.
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JEFFREY R. ANDERSON
max
f Ž x, ? , ¨ g Ž ? . . h9 Ž ? . g Ž ? .
xgV
lim inf min q tª0
g L` Ž Ž 0, T . . ,
fu Ž x, t , ¨ g Ž t . . h9 Ž t .
xgV
)0
and
for 0 - ¨ F M.
THEOREM 2.2. If Ž NU . is satisfied, then the maximal solution, U, of Ž RD . with c ' 0, u 0 ' 0 has U k 0. The two theorems above settle the question of unique solvability for ŽRD. in all cases of power laws except p s m - 1, a ) l 0 . Nonuniqueness of the trivial solution if p s m - 1, a ) l0 can be concluded from the following result, which employs a variation of ŽNU.. THEOREM 2.3. Assume that Ži. and Žii. of Ž NU . are satisfied. If there exists w Ž x . satisfying 0 F w Ž x . F M, w k 0, D f Ž x, t, w Ž x . g Ž t .. q u f Ž x, t, w Ž x . g Ž t .. G 0 on V T , and w s 0 on Ž V .T for some u g w0, 1., and if g 9Ž t . F Ž1 y u . FM Ž t, g Ž t .., then the maximal solution, U, of Ž RD . with c ' 0, u 0 ' 0 has U k 0. Although these nonuniqueness results apply easily to the power law cases, application to other instances of f and f may be somewhat limited. The conditions imposed are essentially just what is needed to construct a subsolution of the form g Ž t . ¨ Ž x, hŽ t .. or g Ž t . w Ž x ., with g Ž0. s 0 and ¨ , w k 0. Such a subsolution allows the conclusion U k 0 for the maximal solution of ŽRD. having c ' 0, u 0 ' 0. 3. PROOF OF THE UNIQUENESS THEOREM Let u denote a solution of ŽRD., and let u k denote the solution of ŽRDk . with the same initial and Dirichlet boundary data. Reversing the integral formulation used to prove u F u k in the previous section, we consider
HV Ž u
k
y u . j Ž x, t . dx
F
t
H0 HV Ž u
k
y u . j s q f Ž x, s, u k . y f Ž x, s, u . D j 4 dx ds
q
t
f Ž x, s, u k . y f Ž x, s, u . j y m Ž k . j 4 dx ds
y
t
f Ž x, s, c q k . y f Ž x, s, c . =j ? n dS x ds
H0 HV H0 H V
HV j Ž x, 0. dx.
qk
NONLINEAR REACTION-DIFFUSION MODEL
489
Choosing j Ž x, t . ' j Ž x . so that D j q l0 j s 0 on V, j s 0 on V, and j G 0, we have
HV Ž u
k
y u . j Ž x, t . dx
F
t H0 HV Ž D t
y
H0 HV
M
q C k . Ž u k y u . y m Ž k . j dx ds
f Ž x, s, c q k . y f Ž x, s, c . =j ? n dS x ds
HV j Ž x . dx,
qk
where C k G < f uŽ x, t, u.< for Ž x, t . g V T and M F u F 5 u k 5 ` . Note here that the ordering u F u k has been employed to ensure f Ž x, t , u k . y f Ž x, t , u . F CM f Ž x, t , u k . y f Ž x, t , u . q Ž DM q C k . Ž u k y u . , even if u k , u F M is not satisfied. Letting k ª 0q, there follows
HV Ž U y u . Ž x, t . j Ž x . dx F Ž C
M
q C1 .
t
H0 HV Ž U y u . Ž x, s . j Ž x . dx ds,
and hence
HV Ž U y u . Ž x, t . j Ž x . dx F 0. Therefore, U y u F 0, and the proof of u ' U is complete.
4. PROOF OF THE NONUNIQUENESS THEOREM As previously shown, if u ' uŽ x, t; c , u 0 . is a subsolution of ŽRD. and U ' UŽ x, t; c , u 0 . is the maximal solution of ŽRD., then u F U. This also follows immediately from the fact that u is a subsolution of ŽRDk ., and subsolutionrsupersolution comparison is known to be true for ŽRDk . when k ) 0 w1x. Hence, u F u k for each k ) 0, which implies u F U. Subsolution comparison for the maximal solution, which is true even in the absence of a uniqueness result for ŽRD., is the key to our proof of nonuniqueness when c ' 0, u 0 ' 0. Formally, a ‘‘classical’’ subsolution of ŽRD., uŽ x, t ., may be found by solving the inequality u t y D f Ž x, t , u . y f Ž x, t , u . F 0,
490
JEFFREY R. ANDERSON
subject to u s 0 on Ž V .T and V = 04 . To construct such a function, we set u Ž x, t . ' g Ž t . ¨ Ž x, h Ž t . . , where g Ž t ., hŽ t . are as introduced in ŽNU.. With
f˜ Ž x, t , ¨ . '
f Ž x, hy1 Ž t . , g ( hy1 Ž t . ¨ . h9( hy1 Ž t . g ( hy1 Ž t .
,
choose ¨ Ž x, t . as the solution of
˜ Ž x, t , ¨ . ¨t s Df ¨ s0
on V T
on Ž V . T
¨ k 0, 0 F ¨ - M
Ž S.
on V = 0 4 .
The assumption ŽNU.Žiii. may now be seen as providing a sufficient condition for problem ŽS. to have a Žlocal. weak solution w1x. Although ¨ Ž x, t . so constructed is only a weak solution, the integral formulation quickly yields that u is a subsolution in the weak sense. Moreover, since ¨ k 0, the inequality u F U establishes U k 0. Hence, the trivial solution of ŽRD. is not unique. There remains only to establish a nonuniqueness result that is applicable to p s m - 1, a ) l0 . Theorem 2.3 is designed for the purpose of handling this exceptional case, and its proof is simply a matter of verifying that uŽ x, t . ' w Ž x . g Ž t . is a classical subsolution of ŽRD.. We only need observe that u t y D f Ž x, t , u . y f Ž x, t , u . s g 9 Ž t . w Ž x . y D f Ž x, t , w Ž x . g Ž t . . y f Ž x, t , w Ž x . g Ž t . . F w Ž x . g 9 Ž t . y Ž 1 y u . FM Ž t , g Ž t . . y D f Ž x, t , w Ž x . g Ž t . . q u f Ž x, t , w Ž x . g Ž t . . F 0. In the case of power laws with p s m - 1, a ) l0 , the assumptions of Theorem 2.3 are satisfied with u s l0ra and DŽ w Ž x .. m q l0 Ž w Ž x .. p s 0 on V, w s 0 on V. Although the question of uniqueness for ŽRD. in the case u 0 k 0 is still left open, it is possible to show the existence of a nontrivial stationary state for the time-independent laws f ' f Ž x, u., f ' f Ž x, u., c ' 0. Specifically, under the conditions ŽNU. Žor the alternative conditions in Theorem
NONLINEAR REACTION-DIFFUSION MODEL
491
2.3., and f Ž?, u. monotone increasing for u ) 0, there exists a nontrivial stationary state, w Ž x ., of u t s D f Ž x, u . q f Ž x, u . us0 u s u0
on V T
on Ž V . T
$
Ž RD.
on V = 0 4 . $
Moreover, if U is the maximal solution of ŽRD. with U s w for t s 0, then $ U ' w. This allows the conclusion w k 0, and the solution of ŽRD. with initial state w Ž x . can be shown to be unique. THEOREM 4.1. Assume Ž NU . is satisfied and f Ž x, u. is monotone increasing in u G 0. Let F Ž x, ? . ' fy1 Ž x, ? . and F Ž x, u. ' f Ž x, F Ž x, u... If lim sup uª`wmax x g V F Ž x, u.ru r x - ` for some 0 F r - 1, then there exists a $ nontri¨ ial stationary solution, w Ž x ., of ŽRD., and UŽ?, t; 0, w . ' w. If, additionally, F Ž x, u.ru is monotone decreasing in u for each x g V, then uŽ?, t; 0, w . ' w. Proof. We begin by constructing w Ž x .. To this end, define ¨ 1Ž x . ' l, and ¨ nŽ x . is the solution of D¨ n q F Ž x, ¨ ny1 . s 0 in V, ¨ n s l on V for n s 2, 3, . . . . By standard monotonicity arguments, l s ¨ 1 F ¨ 2 F ??? F ¨ ny 1 F ¨ n F C. The existence of an upper bound C is guaranteed, since max x g V F Ž x, u. F K 1 q K 2 u r for u G 0 and constants K 1 , K 2 G 0. Hence, ¨ l Ž x . ' lim nª` ¨ nŽ x . exists, and, moreover, ¨ l Ž x . is a solution of D¨ q F Ž x, ¨ . s 0 in V, ¨ s l on V. Now, since ¨ l is monotone decreasing$ in l, w Ž x . ' F Ž x, ¨ Ž x .., where ¨ ' lim l ª 0q ¨ l , is a stationary solution of ŽRD.. Since f Ž x, u. G 0, we have mŽ l . s 0 for $ each l ) 0. It follows that w l Ž x . ' F Ž x, ¨ l Ž x .. is a supersolution of ŽRDk ., the time-independent version of ŽRDk . with c ' 0, for every 0 - k F min x g V F Ž x, l .. Furthermore, we may invoke the maximum principle on the sequence ¨ n4 to conclude ¨ l G ¨ k q Ž l y k . for 0 - k F l. Hence, w l Ž x . s w k Ž x . q Fu Ž x, ? .Ž ¨ l Ž x . y ¨ k Ž x .., which implies w l Ž x . G w k Ž x . q ˆ k, provided ˆ kF Ž l y k .min x g V , k F uF C FuŽ x, u.. $ Therefore, if u ˆk is the solution of ŽRDk . with u ˆk Ž x, 0. s w Ž x . q ˆ k, $ then w F u ˆk F w l. Letting ˆ k ª 0q, we obtain the maximal solution of ŽRD., U ' UŽ?, ? ; 0, w ., and w F U F w l. As l ª 0q, we now have U ' w. Since $ the trivial solution of ŽRD. has been shown to be nonunique, it must follow that w k 0. Furthermore, if u s uŽ?, ? ; 0, w . is any other solution of $ ŽRD., then u F w. Utilizing the weak formulation, with u ˆ ' f Ž?, u. and test function j Ž x, t . ' ¨ Ž x . Ž ¨ is the equilibrium solution constructed
492
JEFFREY R. ANDERSON
above., we have
HV ¨ Ž x .
u Ž x, t . y w Ž x . dx s
t
H0 HV
yuF ˆ Ž x, ¨ . q ¨ F Ž x, uˆ. dx ds G 0.
Therefore, uŽ x, t . y w Ž x . s 0, i.e., u ' w.
5. EXTENSION TO A REACTION-DIFFUSION-CONVECTION MODEL In this final section, we give a possible extension of the uniqueness theorem to a reaction-diffusion model with convection: u t s D f Ž x, t , u . q = ? G Ž x, t , u . q f Ž x, t , u . usc u s u0
on V T
on Ž V . T
Ž RDC.
on V = 0 4 .
Our work here is intended only to exhibit a result that is provable by the methods used herein, and we do not obtain necessary and sufficient conditions in the same sense as for ŽRD.. THEOREM 5.1. that
Assume for each M ) 0 that there exists K M ) 0 such
G Ž x, t , u . y G Ž x, t , ¨ . F K M f Ž x, t , u . y f Ž x, t , ¨ . for all 0 F ¨ F u F M, Ž x, t . g V T . Furthermore, assume that Ž ML. is satisfied. If there exists a nonnegati¨ e function j s j Ž x . with j s 0 on V and D j q K M < =j < q CM j F 0
in V ,
then the solution of Ž RDC . is unique. Of course, this theorem provides a sufficient condition for uniqueness that hinges on the existence of a solution of a differential inequality. It may be difficult, in general, to verify that such a function j is constructible. However, given the existence of j , a maximal solution of ŽRDC. is established w1x and the proof of uniqueness follows our previous work exactly. To address the existence of j , we show that such a function may be constructed if V is a ball of radius R, BR , or if V is an interval in one dimension. In the case that V s BR , we seek j Ž x . s j Ž< x <.. Thus, with
NONLINEAR REACTION-DIFFUSION MODEL
493
r ' < x <, j must satisfy
jr r q
Ny1 r
j r q K M < j r < q CM j F 0.
If CM F Žpr2 R . 2 and K M F Ž N y 1.rR, then j Ž x . ' cosŽp < x
j Ž x . s e K M x r2 sin b Ž R y x . , for x g w0, R x. If b g Ž0, pr2 R . is chosen so that b cotŽ b R . s K M r2, which can be done provided K M r2 - 1rR, and if CM F K M2 r4 q b 2 , then we claim that j 0 q K M < j 9 < q CM j F 0, j ) 0 on V, j ŽyR . s j Ž R . s 0. As j s 0 on V is immediate from the definition of j , we need only verify the differential inequality. Observe that on w0, R x, KM
j 9 Ž x . s e K M x r2 s e K M x r2
KM
½
2
sin b R
b
sin b R y b cos b R cos b x
KM
y sy
sin b Ž R y x . y b cos b Ž R y x .
2
2
ž
K M2 4
cos b R q b sin b R sin b x
5
q b 2 e K M x r2 sin b x,
/
so j 9 - 0 on Ž0, R x with j 9Ž0. s 0. Thus, on w0, R x,
j 0 q K M < j 9 < q CM j F j 0 y KM j 9 q sy q s 0.
sin b R
b
ž
K M2 4
ž
ž
K M2 4
K M2 4
qb2 j
/
q b 2 e K M x r2
/
yK M 2
sin b x q b cos b x
q b 2 e K M x r2 w sin b R cos b x y cos b R sin b x x
/
494
JEFFREY R. ANDERSON
Since j Ž x . s j Žyx . for x g wyR, 0x, it now follows that j 9 G 0 on wyR, 0x, and the differential inequality is satisfied on all of V. We have thus established that the solutions of ŽRDC. are unique if CM and K M can be made sufficiently small for some M ) 0. As in the corresponding result for ŽRD., the smallness requirement of CM places no restriction on 5 u 0 5 ` , 5 c 5 ` . However, in general, the condition on K M will mean that we have uniqueness and comparison of solutions if 5 u 0 5 ` , 5 c 5 ` - M. In the case of power laws, i.e., with G ' ² u n1 , u n 2 , . . . , u n N :, an appropriate rescaling ¨ Ž x, t . ' kuŽ x, ct . removes any condition on the size of 5 u 0 5 ` , 5 c 5 ` . Therefore, for the power law version of ŽRDC., we have established the uniqueness and comparison of nonnegative solutions provided p G min m, 14 Žor a - 0., n i ) m, and V s BR . In one dimension, Gilding has established uniqueness for the nonreactive model Ž a s 0. if n1 ) 0 w5x. This result has been extended to ŽRDC. with a ) 0 only in the case where m, n1 , p G 1 w1x. Hence, the condition n1 ) m required herein represents an improvement on known results if m - 1 Žfast diffusion.. On the other hand, very little, beyond a result of Nanbu for the case a F 0; p, n i , m G 1 w9x, appears to be known regarding the uniqueness of solutions for ŽRDC. when N ) 1. In future work, we plan to investigate necessary and sufficient conditions for the unique solvability of ŽRDC..
REFERENCES 1. J. R. Anderson, Local existence and uniqueness of solutions of degenerate parabolic equations, Comm. Partial Differential Equations 16 Ž1. Ž1991., 105]143. 2. D. G. Aronson, M. G. Crandall, and L. A. Peletier, ‘‘Stabilization of Solutions of a Degenerate Nonlinear Diffusion Problem,’’ University of Wisconsin]Madison Mathematics Research Center, Technical Summary Report 2220, 1981. 3. H. Brezis and M. G. Crandall, Uniqueness of solutions of the initial-value problem for u t y D f Ž u. s 0Ž)., J. Math. Pures Appl. 58 Ž1979., 153]163. 4. J. Filo, On solutions of a perturbed fast diffusion equation, Aplikace Matematiky, 32 Ž5. Ž1987., 364]380. 5. B. H. Gilding, Improved theory for a nonlinear degenerate parabolic equation, Per¨ enuto alla Redazione 17 Ž1986., 165]224. 6. A. S. Kalashnikov, On quasilinear degenerating parabolic equations with singular lowerorder terms and growing initial values. Differentsial’nye Ura¨ neniya 29 Ž6. Ž1993., 999]1009. 7. H. A. Levine and P. E. Sacks, Some existence and nonexistence theorems for solutions of degenerate parabolic equations. J. Differential Equations, 52 Ž2. Ž1984., 135]161. 8. Z. Li and L. A. Peletier, A comparison principle for the porous media equation with absorption, J. Math. Anal. Appl. 165 Ž1992., 457]471. 9. T. Nanbu, Some degenerate nonlinear parabolic equations, Math. Rep. Kyushu Uni¨ . 14 Ž2. Ž1984., 91]110.
228, 483]494 Ž1998.
AY986165
Necessary and Sufficient Conditions for the Unique Solvability of a Nonlinear Reaction-Diffusion Model Jeffrey R. Anderson Department of Mathematics and Statistics, Winona State Uni¨ ersity, Winona, Minnesota 55987 Submitted by John La¨ ery Received December 30, 1997
It has been known for some time that a nonlinear reaction-diffusion model, with Dirichlet boundary conditions, is uniquely solvable if the reaction term satisfies an appropriate Lipschitz condition. However, as recently shown for an absorption model, such a condition is not necessary. We establish a uniqueness result which, in the case of reaction and diffusion governed by power laws, is in fact both necessary and sufficient for the unique solvability of the model. The improvement that is needed on the above-mentioned Lipschitz condition occurs in the so-called fast diffusion model. Q 1998 Academic Press
1. INTRODUCTION The goal of this investigation is to obtain conditions that guarantee unique solvability of the nonlinear reaction-diffusion model, u t s D f Ž x, t , u . q f Ž x, t , u . usc u s u0
on Ž V . T
on V T
Ž RD.
on V = 0 4 ,
and to study the extent to which these are necessary. Here, V ; R N is a bounded domain with C 2 boundary, V T ' V = Ž0, T ., and c G 0, u 0 G 0, with u 0 g L`Ž V ., c g L`ŽŽ0, T ... We shall assume f Ž?, ? , 0. s 0 and, hence, consider only nonnegative solutions of the model. The parabolicity condition, fu ) 0 for u ) 0, and smoothness conditions required of f and f 483 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
484
JEFFREY R. ANDERSON
are exactly those that are sufficient for the construction of a local weak solution. In the case of power laws, f s u m , f s au p , these allow all positive powers m, p ) 0. If f is assumed to be Žlocally. Lipschitz continuous in u, then the uniqueness of solutions of ŽRD. has appeared in many articles, e.g., w1]4, 7x. In the case of an ‘‘absorption law,’’ i.e., a F 0, it has been shown that Lipschitz continuity is no longer necessary w8x. Moreover, as will be established, the Lipschitz condition on f is not necessary, even in the reactive case, a ) 0. Toward developing a uniqueness result for ŽRD. that is both necessary and sufficient in the instance of power laws, we propose the following modified Lipschitz condition on f : There exist positive constants CM , DM such that f Ž x, t , u . y f Ž x, t , ¨ . F CM f Ž x, t , u . y f Ž x, t , ¨ . q DM w u y ¨ x
Ž ML .
for every 0 F ¨ F u F M. We show that ŽML. is sufficient for the uniqueness of all nonnegative solutions of ŽRD. provided CM F l 0 for some M ) 0, and l 0 is the smallest Dirichlet eigenvalue for V. Such a condition is satisfied by the power laws if a F 0 or p G min m, 14 Žand a F l0 in the situation p s m - 1.. The remaining cases of a, m, p are contained in two nonuniqueness results for ŽRD., in which we prove that the trivial solution is not unique. Thus, ŽML. with CM F l 0 is both necessary and sufficient for the unique solvability of the power law version of ŽRD.. This is our main result. It can be seen that ŽML. differs significantly from the standard Lipschitz condition in a model of ‘‘fast’’ diffusion, e.g., 0 - m - 1. In the next section we give a precise definition of a weak Žlocal. solution of ŽRD. and state our main results on uniqueness and nonuniqueness for such solutions. A maximal weak solution is then constructed via monotone limits of solutions to suitably ‘‘regularized’’ models. Although the construction of such a solution is not new Že.g., see w1, 2, 6x., it is key to the development of uniqueness and nonuniqueness for ŽRD.. Having constructed the maximal solution, U ' UŽ x, t; c , u 0 ., we have u F U for any other solution, u s uŽ x, t; c , u 0 .. In Section 3, we prove the unique solvability of ŽRD. by establishing that the reverse inequality is also true. A benefit of the monotone construction of U is that once solutions of ŽRD. are known to be unique, solution and subsolution comparison principles follow immediately. As our method makes use of the key fact U G u, this work does not appear to allow development of a general supersolution comparison result. In Section 4, we utilize the construction of U and show that, under suitable conditions, a subsolution argument can be fashioned that yields
NONLINEAR REACTION-DIFFUSION MODEL
485
UŽ?, ? ; 0, 0. k 0. Thus, the trivial solution is not unique. Whether or not solutions are unique if u 0 k 0, c ' 0 is yet an open problem. On the other hand, we give instances in which ŽRD. is not uniquely solvable for trivial initial and Dirichlet data, has a nontrivial equilibrium state, w, and, yet, uŽ?, ? ; 0, w . s w for any solution, u, of ŽRD.. That is, we have the uniqueness of solutions if c ' 0 and u 0 ' w. In the final section, we present a possible generalization of this work to a reaction-diffusion model with convection. However, in the presence of convection, a complete study of necessary and sufficient conditions for the unique solvability of the model is left for a future study.
2. MAXIMAL SOLUTION OF ŽRD. AND STATEMENT OF THE MAIN RESULTS To construct a maximal solution of ŽRD., assume Ži. V T = R. Žii. Žiii. Živ.
f, f , D x f , and the components of =x f are all continuous on The partial derivatives fu , f u are continuous on V T = R R 04 . fu ) 0 for u ) 0. f Ž x, t, 0. s f Ž x, t, 0. s 0.
Here, D x and =x are used to denote the Laplacian and gradient operators, respectively, each taken with respect to the x variables only. ŽContinuity in x and t can actually be replaced with a weaker condition, such as inclusion in L` , and the construction used herein is still valid w1x.. We also assume that V is a bounded domain in R N with C 2 boundary, i.e., V is locally representable by twice continuously differentiable functions. The outward pointing unit normal to V is denoted by n. As the differential equation is potentially of degenerate parabolic type, we do not expect classical solutions of ŽRD. in general. The notion of a weak Žlocal. solution adopted here is the same as in previous work w1x. To state the definition, let the class of nonnegative test functions be defined as T ' j : j t , D j , < =j < g L2 Ž V T . , j G 0, and j < Ž V . T s 0 4 . DEFINITION 2.1. A function uŽ x, t . is called a subsolution Žsupersolution. of ŽRD. on V T if the following all hold: Ži. Žii.
u g L`Ž V T .. u F ŽG. u 0 on V = 04 .
486
JEFFREY R. ANDERSON
Žiii. u F ŽG. c on Ž V .T . Živ. For every t g w0, T x and every j g T ,
HV u Ž x, t . j Ž x, t . dx F Ž G. HV u Ž x . j Ž x, 0. dx 0
t
H0 H V f Ž x, s, c Ž s . . =j ? n dS
y
t
H0 HV u j
q
s
x
ds
q f Ž x, s, u . D j q f Ž x, s, u . j 4 dx ds.
A function uŽ x, t . is called a Žlocal. solution of ŽRD. if it is both a subsolution and a supersolution of ŽRD. on V T for some T ) 0. To construct such a solution, define m Ž k . ' min
½
min
V T= w0, 2 k x
wDxf q f x, 0
5
and let u k denote the solution of u t s D f Ž x, t , u . q f Ž x, t , u . y m Ž k . uscqk u s u0 q k
on V T
on Ž V . T
Ž RDk .
on V = 0 4 .
T ) 0 may be chosen sufficiently small in such a way that there exists a solution of ŽRDk . on V T for every 0 - k F 1, and 5 u k 5 ` is bounded independently of k. Furthermore, k F u k F u l for 0 - k F l, and a subsolutionrsupersolution comparison theory holds for ŽRDk . w1x. As u k is monotone in k, we may define U ' lim k ª 0q u k , and it is easy to see that U is a solution of ŽRD.. Furthermore, if u is a solution of ŽRD. such that u s U on Ž V .T and on V = 04 , then
HV Ž u y u s
k
. j Ž x, t . dx
t
H0 HV Ž u y u
k
. j s q f Ž x, s, u . y f Ž x, s, u k . D j 4 dx ds
q
t
f Ž x, s, u . y f Ž x, s, u k . j q m Ž k . j 4 dx ds
y
t
f Ž x, s, c . y f Ž x, s, c q k . =j ? n dS x ds
H0 HV H0 H V
HV j Ž x, 0. dx.
yk
NONLINEAR REACTION-DIFFUSION MODEL
487
With F k and Fk defined so that
Ž u y u k . F k s f Ž x, s, u . y f Ž x, s, u k . and
Ž u y u k . Fk s f Ž x, s, u . y f Ž x, s, u k . , we have
HV Ž u y u
t
k
. j Ž x, t . dx F H
0
HV Ž u y u
k
. j s q F k D j q Fk j 4 dx ds.
Note here the use of yD j ? n G 0 on Ž V .T and ymŽ k . G 0 to derive this inequality. The choice of j according to j G 0, j s 0 on Ž V .T , and j s q F k D j q Fk j s 0 now yields u F u k . ŽIf necessary, F k and Fk may be smoothed out appropriately to obtain j , and passage to the limit as the smoothings are removed may be justified w1, 2x.. If u is a subsolution of ŽRD., the above argument shows that u F u k is generally true if the inequality holds on V = 04 j Ž V .T . Thus, U is the maximal solution of ŽRD., and this solution satisfies a subsolution comparison theory. To establish uniqueness for solutions of ŽRD., it only remains to prove that the reverse inequality is also true. Our result in this direction is the following. THEOREM 2.1. If Ž ML. is satisfied and CM F l 0 for some M ) 0, where l0 ) 0 is the smallest Dirichlet eigen¨ alue for V, then the Ž nonnegati¨ e . solution of Ž RD . is unique. Furthermore, if c 1 G c 2 G 0, u 0 G ¨ 0 G 0, u s uŽ x, t; c 1 , u 0 . is a solution of Ž RD ., and ¨ s ¨ Ž x, t; c 2 , ¨ 0 . is a subsolution of Ž RD ., then u G ¨ . Toward an investigation of multiple solutions for ŽRD., we introduce the following rather elaborate condition, which may be applied to the power laws. ŽHere the label ŽNU. is used to denote a sufficient condition for ‘‘nonuniqueness.’’.
Ž i.
There exists FM Ž t , k . such that f Ž x, t , k¨ . G ¨ FM Ž t , k .
Ž ii .
for x g V , t g Ž 0, T . , 0 - ¨ F M, k G 0. There exists g Ž t . such that g Ž 0 . s 0 with 0 - g 9 Ž t . F FM Ž t , g Ž t . . for t g Ž 0, T . .
Ž iii .
There exists h Ž t . such that h Ž 0 . s 0, h9 Ž t . ) 0 for t g Ž 0, T . ,
Ž NU.
488
JEFFREY R. ANDERSON
max
f Ž x, ? , ¨ g Ž ? . . h9 Ž ? . g Ž ? .
xgV
lim inf min q tª0
g L` Ž Ž 0, T . . ,
fu Ž x, t , ¨ g Ž t . . h9 Ž t .
xgV
)0
and
for 0 - ¨ F M.
THEOREM 2.2. If Ž NU . is satisfied, then the maximal solution, U, of Ž RD . with c ' 0, u 0 ' 0 has U k 0. The two theorems above settle the question of unique solvability for ŽRD. in all cases of power laws except p s m - 1, a ) l 0 . Nonuniqueness of the trivial solution if p s m - 1, a ) l0 can be concluded from the following result, which employs a variation of ŽNU.. THEOREM 2.3. Assume that Ži. and Žii. of Ž NU . are satisfied. If there exists w Ž x . satisfying 0 F w Ž x . F M, w k 0, D f Ž x, t, w Ž x . g Ž t .. q u f Ž x, t, w Ž x . g Ž t .. G 0 on V T , and w s 0 on Ž V .T for some u g w0, 1., and if g 9Ž t . F Ž1 y u . FM Ž t, g Ž t .., then the maximal solution, U, of Ž RD . with c ' 0, u 0 ' 0 has U k 0. Although these nonuniqueness results apply easily to the power law cases, application to other instances of f and f may be somewhat limited. The conditions imposed are essentially just what is needed to construct a subsolution of the form g Ž t . ¨ Ž x, hŽ t .. or g Ž t . w Ž x ., with g Ž0. s 0 and ¨ , w k 0. Such a subsolution allows the conclusion U k 0 for the maximal solution of ŽRD. having c ' 0, u 0 ' 0. 3. PROOF OF THE UNIQUENESS THEOREM Let u denote a solution of ŽRD., and let u k denote the solution of ŽRDk . with the same initial and Dirichlet boundary data. Reversing the integral formulation used to prove u F u k in the previous section, we consider
HV Ž u
k
y u . j Ž x, t . dx
F
t
H0 HV Ž u
k
y u . j s q f Ž x, s, u k . y f Ž x, s, u . D j 4 dx ds
q
t
f Ž x, s, u k . y f Ž x, s, u . j y m Ž k . j 4 dx ds
y
t
f Ž x, s, c q k . y f Ž x, s, c . =j ? n dS x ds
H0 HV H0 H V
HV j Ž x, 0. dx.
qk
NONLINEAR REACTION-DIFFUSION MODEL
489
Choosing j Ž x, t . ' j Ž x . so that D j q l0 j s 0 on V, j s 0 on V, and j G 0, we have
HV Ž u
k
y u . j Ž x, t . dx
F
t H0 HV Ž D t
y
H0 HV
M
q C k . Ž u k y u . y m Ž k . j dx ds
f Ž x, s, c q k . y f Ž x, s, c . =j ? n dS x ds
HV j Ž x . dx,
qk
where C k G < f uŽ x, t, u.< for Ž x, t . g V T and M F u F 5 u k 5 ` . Note here that the ordering u F u k has been employed to ensure f Ž x, t , u k . y f Ž x, t , u . F CM f Ž x, t , u k . y f Ž x, t , u . q Ž DM q C k . Ž u k y u . , even if u k , u F M is not satisfied. Letting k ª 0q, there follows
HV Ž U y u . Ž x, t . j Ž x . dx F Ž C
M
q C1 .
t
H0 HV Ž U y u . Ž x, s . j Ž x . dx ds,
and hence
HV Ž U y u . Ž x, t . j Ž x . dx F 0. Therefore, U y u F 0, and the proof of u ' U is complete.
4. PROOF OF THE NONUNIQUENESS THEOREM As previously shown, if u ' uŽ x, t; c , u 0 . is a subsolution of ŽRD. and U ' UŽ x, t; c , u 0 . is the maximal solution of ŽRD., then u F U. This also follows immediately from the fact that u is a subsolution of ŽRDk ., and subsolutionrsupersolution comparison is known to be true for ŽRDk . when k ) 0 w1x. Hence, u F u k for each k ) 0, which implies u F U. Subsolution comparison for the maximal solution, which is true even in the absence of a uniqueness result for ŽRD., is the key to our proof of nonuniqueness when c ' 0, u 0 ' 0. Formally, a ‘‘classical’’ subsolution of ŽRD., uŽ x, t ., may be found by solving the inequality u t y D f Ž x, t , u . y f Ž x, t , u . F 0,
490
JEFFREY R. ANDERSON
subject to u s 0 on Ž V .T and V = 04 . To construct such a function, we set u Ž x, t . ' g Ž t . ¨ Ž x, h Ž t . . , where g Ž t ., hŽ t . are as introduced in ŽNU.. With
f˜ Ž x, t , ¨ . '
f Ž x, hy1 Ž t . , g ( hy1 Ž t . ¨ . h9( hy1 Ž t . g ( hy1 Ž t .
,
choose ¨ Ž x, t . as the solution of
˜ Ž x, t , ¨ . ¨t s Df ¨ s0
on V T
on Ž V . T
¨ k 0, 0 F ¨ - M
Ž S.
on V = 0 4 .
The assumption ŽNU.Žiii. may now be seen as providing a sufficient condition for problem ŽS. to have a Žlocal. weak solution w1x. Although ¨ Ž x, t . so constructed is only a weak solution, the integral formulation quickly yields that u is a subsolution in the weak sense. Moreover, since ¨ k 0, the inequality u F U establishes U k 0. Hence, the trivial solution of ŽRD. is not unique. There remains only to establish a nonuniqueness result that is applicable to p s m - 1, a ) l0 . Theorem 2.3 is designed for the purpose of handling this exceptional case, and its proof is simply a matter of verifying that uŽ x, t . ' w Ž x . g Ž t . is a classical subsolution of ŽRD.. We only need observe that u t y D f Ž x, t , u . y f Ž x, t , u . s g 9 Ž t . w Ž x . y D f Ž x, t , w Ž x . g Ž t . . y f Ž x, t , w Ž x . g Ž t . . F w Ž x . g 9 Ž t . y Ž 1 y u . FM Ž t , g Ž t . . y D f Ž x, t , w Ž x . g Ž t . . q u f Ž x, t , w Ž x . g Ž t . . F 0. In the case of power laws with p s m - 1, a ) l0 , the assumptions of Theorem 2.3 are satisfied with u s l0ra and DŽ w Ž x .. m q l0 Ž w Ž x .. p s 0 on V, w s 0 on V. Although the question of uniqueness for ŽRD. in the case u 0 k 0 is still left open, it is possible to show the existence of a nontrivial stationary state for the time-independent laws f ' f Ž x, u., f ' f Ž x, u., c ' 0. Specifically, under the conditions ŽNU. Žor the alternative conditions in Theorem
NONLINEAR REACTION-DIFFUSION MODEL
491
2.3., and f Ž?, u. monotone increasing for u ) 0, there exists a nontrivial stationary state, w Ž x ., of u t s D f Ž x, u . q f Ž x, u . us0 u s u0
on V T
on Ž V . T
$
Ž RD.
on V = 0 4 . $
Moreover, if U is the maximal solution of ŽRD. with U s w for t s 0, then $ U ' w. This allows the conclusion w k 0, and the solution of ŽRD. with initial state w Ž x . can be shown to be unique. THEOREM 4.1. Assume Ž NU . is satisfied and f Ž x, u. is monotone increasing in u G 0. Let F Ž x, ? . ' fy1 Ž x, ? . and F Ž x, u. ' f Ž x, F Ž x, u... If lim sup uª`wmax x g V F Ž x, u.ru r x - ` for some 0 F r - 1, then there exists a $ nontri¨ ial stationary solution, w Ž x ., of ŽRD., and UŽ?, t; 0, w . ' w. If, additionally, F Ž x, u.ru is monotone decreasing in u for each x g V, then uŽ?, t; 0, w . ' w. Proof. We begin by constructing w Ž x .. To this end, define ¨ 1Ž x . ' l, and ¨ nŽ x . is the solution of D¨ n q F Ž x, ¨ ny1 . s 0 in V, ¨ n s l on V for n s 2, 3, . . . . By standard monotonicity arguments, l s ¨ 1 F ¨ 2 F ??? F ¨ ny 1 F ¨ n F C. The existence of an upper bound C is guaranteed, since max x g V F Ž x, u. F K 1 q K 2 u r for u G 0 and constants K 1 , K 2 G 0. Hence, ¨ l Ž x . ' lim nª` ¨ nŽ x . exists, and, moreover, ¨ l Ž x . is a solution of D¨ q F Ž x, ¨ . s 0 in V, ¨ s l on V. Now, since ¨ l is monotone decreasing$ in l, w Ž x . ' F Ž x, ¨ Ž x .., where ¨ ' lim l ª 0q ¨ l , is a stationary solution of ŽRD.. Since f Ž x, u. G 0, we have mŽ l . s 0 for $ each l ) 0. It follows that w l Ž x . ' F Ž x, ¨ l Ž x .. is a supersolution of ŽRDk ., the time-independent version of ŽRDk . with c ' 0, for every 0 - k F min x g V F Ž x, l .. Furthermore, we may invoke the maximum principle on the sequence ¨ n4 to conclude ¨ l G ¨ k q Ž l y k . for 0 - k F l. Hence, w l Ž x . s w k Ž x . q Fu Ž x, ? .Ž ¨ l Ž x . y ¨ k Ž x .., which implies w l Ž x . G w k Ž x . q ˆ k, provided ˆ kF Ž l y k .min x g V , k F uF C FuŽ x, u.. $ Therefore, if u ˆk is the solution of ŽRDk . with u ˆk Ž x, 0. s w Ž x . q ˆ k, $ then w F u ˆk F w l. Letting ˆ k ª 0q, we obtain the maximal solution of ŽRD., U ' UŽ?, ? ; 0, w ., and w F U F w l. As l ª 0q, we now have U ' w. Since $ the trivial solution of ŽRD. has been shown to be nonunique, it must follow that w k 0. Furthermore, if u s uŽ?, ? ; 0, w . is any other solution of $ ŽRD., then u F w. Utilizing the weak formulation, with u ˆ ' f Ž?, u. and test function j Ž x, t . ' ¨ Ž x . Ž ¨ is the equilibrium solution constructed
492
JEFFREY R. ANDERSON
above., we have
HV ¨ Ž x .
u Ž x, t . y w Ž x . dx s
t
H0 HV
yuF ˆ Ž x, ¨ . q ¨ F Ž x, uˆ. dx ds G 0.
Therefore, uŽ x, t . y w Ž x . s 0, i.e., u ' w.
5. EXTENSION TO A REACTION-DIFFUSION-CONVECTION MODEL In this final section, we give a possible extension of the uniqueness theorem to a reaction-diffusion model with convection: u t s D f Ž x, t , u . q = ? G Ž x, t , u . q f Ž x, t , u . usc u s u0
on V T
on Ž V . T
Ž RDC.
on V = 0 4 .
Our work here is intended only to exhibit a result that is provable by the methods used herein, and we do not obtain necessary and sufficient conditions in the same sense as for ŽRD.. THEOREM 5.1. that
Assume for each M ) 0 that there exists K M ) 0 such
G Ž x, t , u . y G Ž x, t , ¨ . F K M f Ž x, t , u . y f Ž x, t , ¨ . for all 0 F ¨ F u F M, Ž x, t . g V T . Furthermore, assume that Ž ML. is satisfied. If there exists a nonnegati¨ e function j s j Ž x . with j s 0 on V and D j q K M < =j < q CM j F 0
in V ,
then the solution of Ž RDC . is unique. Of course, this theorem provides a sufficient condition for uniqueness that hinges on the existence of a solution of a differential inequality. It may be difficult, in general, to verify that such a function j is constructible. However, given the existence of j , a maximal solution of ŽRDC. is established w1x and the proof of uniqueness follows our previous work exactly. To address the existence of j , we show that such a function may be constructed if V is a ball of radius R, BR , or if V is an interval in one dimension. In the case that V s BR , we seek j Ž x . s j Ž< x <.. Thus, with
NONLINEAR REACTION-DIFFUSION MODEL
493
r ' < x <, j must satisfy
jr r q
Ny1 r
j r q K M < j r < q CM j F 0.
If CM F Žpr2 R . 2 and K M F Ž N y 1.rR, then j Ž x . ' cosŽp < x
j Ž x . s e K M x r2 sin b Ž R y x . , for x g w0, R x. If b g Ž0, pr2 R . is chosen so that b cotŽ b R . s K M r2, which can be done provided K M r2 - 1rR, and if CM F K M2 r4 q b 2 , then we claim that j 0 q K M < j 9 < q CM j F 0, j ) 0 on V, j ŽyR . s j Ž R . s 0. As j s 0 on V is immediate from the definition of j , we need only verify the differential inequality. Observe that on w0, R x, KM
j 9 Ž x . s e K M x r2 s e K M x r2
KM
½
2
sin b R
b
sin b R y b cos b R cos b x
KM
y sy
sin b Ž R y x . y b cos b Ž R y x .
2
2
ž
K M2 4
cos b R q b sin b R sin b x
5
q b 2 e K M x r2 sin b x,
/
so j 9 - 0 on Ž0, R x with j 9Ž0. s 0. Thus, on w0, R x,
j 0 q K M < j 9 < q CM j F j 0 y KM j 9 q sy q s 0.
sin b R
b
ž
K M2 4
ž
ž
K M2 4
K M2 4
qb2 j
/
q b 2 e K M x r2
/
yK M 2
sin b x q b cos b x
q b 2 e K M x r2 w sin b R cos b x y cos b R sin b x x
/
494
JEFFREY R. ANDERSON
Since j Ž x . s j Žyx . for x g wyR, 0x, it now follows that j 9 G 0 on wyR, 0x, and the differential inequality is satisfied on all of V. We have thus established that the solutions of ŽRDC. are unique if CM and K M can be made sufficiently small for some M ) 0. As in the corresponding result for ŽRD., the smallness requirement of CM places no restriction on 5 u 0 5 ` , 5 c 5 ` . However, in general, the condition on K M will mean that we have uniqueness and comparison of solutions if 5 u 0 5 ` , 5 c 5 ` - M. In the case of power laws, i.e., with G ' ² u n1 , u n 2 , . . . , u n N :, an appropriate rescaling ¨ Ž x, t . ' kuŽ x, ct . removes any condition on the size of 5 u 0 5 ` , 5 c 5 ` . Therefore, for the power law version of ŽRDC., we have established the uniqueness and comparison of nonnegative solutions provided p G min m, 14 Žor a - 0., n i ) m, and V s BR . In one dimension, Gilding has established uniqueness for the nonreactive model Ž a s 0. if n1 ) 0 w5x. This result has been extended to ŽRDC. with a ) 0 only in the case where m, n1 , p G 1 w1x. Hence, the condition n1 ) m required herein represents an improvement on known results if m - 1 Žfast diffusion.. On the other hand, very little, beyond a result of Nanbu for the case a F 0; p, n i , m G 1 w9x, appears to be known regarding the uniqueness of solutions for ŽRDC. when N ) 1. In future work, we plan to investigate necessary and sufficient conditions for the unique solvability of ŽRDC..
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