EXISTENCE OF SOLUTIONS FOR THE

Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 9, 1997, 279–296

EXISTENCE OF SOLUTIONS FOR THE DISCRETE COAGULATION-FRAGMENTATION MODEL WITH DIFFUSION

Dariusz Wrzosek

Dedicated to Olga Ladyzhenskaya

1. Introduction We consider the following infinite system of reaction-diffusion equations: ∞ ∞ X X ∂u1 = d1 ∆u1 − u1 a1j uj + b1j u1+j , ∂t j=1 j=1 i−1

(1.1)

∂ui 1X = di ∆ui + (ai−j,j ui−j uj − bi−j,j ui ) ∂t 2 j=1 − ui

∞ X

aij uj +

j=1

∞ X

bij ui+j ,

i = 2, 3, . . . ,

j=1

on ΩT = Ω × (0, T ), subject to the initial condition (1.2)

ui (0, x) = Ui (x)

for x ∈ Ω, i = 1, 2, . . .

and Neumann boundary condition (1.3)

∂ui = 0 on ∂Ω × (0, T ), i = 1, 2, . . . , ∂ν

1991 Mathematics Subject Classification. 35K57, 92E20. The author was supported by the French–Polish project no. 558 and by Polish KBN grant no. 2 PO3A 065 08. c

1997 Juliusz Schauder Center for Nonlinear Studies

279

280

D. Wrzosek

where Ω ⊂ Rn is a bounded domain with smooth boundary, ν is an outward normal vector of Ω, and Ui ∈ L∞ (Ω), i = 1, 2, . . . , are given nonnegative functions. We will refer to problem (1.1)–(1.3) as problem (P). The system (1.1) is a generalization of the discrete coagulation-fragmentation model which describes the dynamics of cluster growth. Such models arise in polymer science [9, 10, 16, 17], atmospheric physics [8] and colloidal chemistry [15], to give a few examples. In models analyzed in this paper the clusters are assumed to be discrete, i.e. they consist of a finite number of smaller particles (resp. polymers and monomers in polymer physics). The variable ui represents the concentration of i-clusters, that is, clusters consisting of i identical elementary particles. The coagulation rates aij and fragmentation rates bij are nonnegative constants such that aij = aji and bij = bji , and di are the diffusion coefficients, di > 0, i = 1, 2, . . . The coefficient aij represents reaction in which an (i + j)-cluster is formed from an i-cluster and a j-cluster, whereas bij is due to break-up of an (i+j)-cluster into i- and j-clusters. The first two terms in the ith equation describe the rate of change of i-clusters caused by coagulation of smaller clusters and by fragmentation. The last two terms represent the interaction of i-clusters with themselves and all others and break-up of larger clusters into i-clusters. Since the size of clusters is not limited a priori , an infinite number of variables is introduced. A model taking into account the diffusion of clusters (in absence of fragmentation) was given in [7]. Some properties of such a model are shown in [14]. We generalize some results from the previous paper [4], where existence and uniqueness of solutions and mass conservation are studied under the hypothesis that aij = ri rj , bij = 0, ri = O(i), i, j ≥ 1. After the manuscript of this article was accomplished the author learned about results obtained by Collet and Poupaud in [5]. They prove the existence of global-in-time solutions only in the case of equal diffusion coefficients in each equation and for a fairly restricted range of coagulation and fragmentation coefficients. For a physical interpretation of the coefficients in the model we refer the reader to [2, 7, 8, 9, 16]. A survey of different variants of the model and numerical methods for the case without diffusion can be found in [6]. We assume two physically meaningful growth conditions of the coagulation coefficients, namely (H1) aij ≤ Aij for i, j ≥ 1, A > 0, and (H2) aij ≤ A(i + j) for i, j ≥ 1, A > 0. Many properties of the coagulation-fragmentation model without diffusion are proved in [2] under the hypothesis (H2).

281

Discrete Coagulation-Fragmentation Model

The hypothesis (H2) is considered separately because in this case solutions to problem (P) preserve for all t ≥ 0 the total mass M0 contained in the initial distribution, i.e. (1.4)

∞ Z ∞ Z X X i ui (x, t) dx = i Ui (x) dx = M0 i=1

Ω

i=1

for t ≥ 0.

Ω

From now on we assume that M0 < ∞. It is known that in the case aij = ri rj ,

ri = iα ,

α > 1/2,

bij = 0,

di = d > 0,

i, j ≥ 1,

(1.4) holds only locally in time. This is related to the sol-gel transition (see [9, 10, 16, 17]) for more details. Similarly to the previous paper [4] solutions of problem (P) are constructed as limits of solutions of finite systems. The paper is organized as follows. In Section 2 we prove some uniform bounds for solutions of finite systems approximating the original one. In Section 3 problem (P) is studied under the assumption that the diffusion coefficients di are arbitrary positive constants. In comparison with the model without diffusion [2], in order to show existence of global-in-time solutions we need the following additional restriction on the fragmentation coefficients: (H3) For each i ≥ 1 there exists γi > 0 such that bi,j−i ≤ γi aij

for j ≥ i + 1.

In this case we are able to prove the existence of global-in-time solutions satisfying ∞ Z X i ui (x, t) dx ≤ M0 . i=1

Ω

In Section 4 we assume that: (H4) there exist M ≥ 1 and d > 0 such that di = d for i ≥ M . This enables us to prove the existence of solutions under more general assumptions on the reaction rate coefficients and also to show (1.4). If the diffusion coefficients are the same in each equation the existence of solutions is proved without any growth condition on bij (see Theorem 4.2). We shall denote by L the L2 -realization of the Laplace operator subject to homogeneous Neumann boundary condition: Lu = ∆u

with D(L) = {u : u ∈ H 2 (Ω), ∂u/∂ν = 0},

and Ωt stands for Ω × (0, t).

282

D. Wrzosek

2. Approximation of solutions Solutions of the infinite system are constructed in the process of approximation by the following systems (PN ) of 2N equations defined for any integer N ≥ 2: N N −1 X X ∂uN 1 N N = d1 ∆uN − u a u + b1j uN 1j 1 1 j 1+j , ∂t i=1 j=1

.. . i−1

(2.1)

1X ∂uN i N N = di ∆uN (ai−j,j uN i + i−j uj − bi−j,j ui ) ∂t 2 j=1 −

N X

N aij uN i uj +

j=1

N −i X

bij uN i+j

for i = 2, . . . , N,

j=1

N 1 X ∂uN i N = di ∆uN aj,i−j uN i + j ui−j ∂t 2

for N + 1 ≤ i ≤ 2N,

j=i−N

subject to boundary and initial conditions as in (1.2), (1.3). Notice that this system corresponds to the first 2N equations of the system (1.1) where aij = 0 for i > N or j > N

and bij = 0

for i + j > N.

In this section we prove some properties of (2.1) which will be useful for further analysis. Proposition 2.1. For any N ≥ 2 the system (PN ) has a unique nonnegative 2N solution {uN k }k=1 defined on Ω × (0, Tmax ): for k = 1, . . . , 2N , 2 uN k ∈ C([0, Tmax [; L (Ω))

uN k uN k uN k

∈ ∈ ∈

and satisfies (1.2),

∞ L∞ loc ([0, Tmax [; L (Ω)), 1,∞ Wloc (]0, Tmax [; L2 (Ω)), L∞ loc (]0, Tmax [; D(L)),

and the equations of (PN ) are satisfied a.e. in Ω × (0, Tmax ). Proof. The system (PN ) can be rewritten in the form ∂uN i N N N N N = di LuN i + fi (u1 , . . . , u2N ) + ui gi (u1 , . . . , u2N ), ∂t

i = 1, . . . , 2N,

where Li is the infinitesimal generator of an analytic semigroup of bounded linear operators on L2 (Ω), and fi , gi are locally Lipschitz continuous functions on R2N with fi ≥ 0 on R2N + . By the general theory of reaction-diffusion equations (see

283


e.g. [11], [13] or [12]) there exists a unique nonnegative, local-in-time solution of (PN ). Multiplying the ith equation in (2.1) by a number gi and then summing them up we obtain (2.2)

2N X i=1

2N

gi

X ∂uN i − gi di ∆uN i ∂t i=1 1 X N (gi+j − gi − gj )aij uN = i uj 2 1≤i,j≤N

−

1 2

X

(gi+j − gi − gj )bij uN i+j

on Ω × (0, Tmax ).

1 0.

j=i−N

Proof. In the first equation of (PN ) take the test function φ1 = (u1 − k1 )+ |[0,t] ,

284

D. Wrzosek

where k1 = max{γ1 , kU1 k∞ } and 0 ≤ t < Tm where Tm is the maximal existence time. Integrating yields Z Z 1 2 2 k(uN (·, t) − k ) k + d |∇(uN 1 + 2 1 1 1 − k1 )+ | 2 Ωt Z Z Z Z X N 2 N N = −a11 (uN − k ) − (a1j uN 1 + 1 1 − b1j−1 )uj (u1 − k1 )+ ≤ 0 Ωt

Ωt j=2

where the last inequality follows from (H3) and from the nonnegativity of solutions. Thus, we have kuN 1 (·, t)kL∞ (Ω) ≤ k1

for 0 ≤ t < Tm ,

uniformly with respect to N. Let kuN j kL∞ (Ωt ) ≤ Kj for 1 ≤ j ≤ i − 1. We shall show that kuN i kL∞ (Ωt ) ≤ Ki ,

(2.7)

where Ki is given by (2.6). To this end let us test the ith equation with the function p−1 φi = p(uN i − ki )+ |[0,t] , where p ≥ 2 and t < Tm . We then obtain, taking ki ≥ kUi kL∞ (Ω) , Z Z p−2 p 2 N (2.8) k(uN (t) − k ) k + d p(p − 1) |∇(uN i + i i p i − ki )+ | (ui − ki )+ Ωt

X i−1

Z Z ≤ Ωt

ai−j,j Ki−j Kj −

j=1

i−1 X

bi−j,j (uN i

− ki )+ −

aii (uN i

−

ki )2+

j=1

p−1 × p(uN i − ki )+ dx dt

X i−1

Z Z − Ωt

N aij uN j ui

+

j=1

N X

(aij uN i

−

bi,j−i )uN j

j=i+1

p−1 × p(uN i − ki )+ dx dt =: I1 − I2 .

By (H3), I2 ≥ 0 if ki ≥ γi . Using the Young inequality “with ε” we find 1−p X p Z Z i−1 p p aij Ki Kj ε1−p |Ω|t + ε (uN I1 ≤ pp−1 i − ki )+ p−1 Ω t j=1 −p

X i−1

Z Z

p (uN i − ki )+ + |Ω|t − paii

bi−j,j Ωt

j=1

Z Z

p (uN i − ki )+ .

Ωt

Pi−1 Taking ε = p( j=1 bi−j,j + aii ) yields (2.9)

I1 ≤

X i−1 j=1

aij Ki Kj

p X i−1 j=1

1−p bi−j,j + aii

+ 1 |Ω|Tm

285


and finally from (2.8) and (2.9) it follows that Z (2.10)

sup lim sup p≥2

t∈[0,Tm ]

1/p p (uN (x, t) − k ) dx i i

Ω

≤

X i−1

aij Ki Kj

X i−1

j=1

−1 bi−j,j + aii

+ 1,

j=1

which yields (2.7). Therefore, solutions can be prolonged for all t > 0, which implies (2.5) and (2.6). The last statement of the theorem follows immediately from the maximum principle. In Lemma 2.3, below, which will be used in Section 4, we shall use the following formula: for M < N ,

(2.11)

2N X ∂uN i − gi di ∆uN i ∂t i=M i=M 1 X N = (gi+j − gi − gj )aij uN i uj 2 2N X

gi

M ≤i,j≤N

X

+

1 2

N (gi+j − gj )aij uN i uj +

1≤i 0 and a constant C1 (T ) such that

X

2N 2 N

(2.12) ≤ C1 (T ) for T ∈ [0, Tm [, i ui (·, t) sup

t∈[0,T ]

(2.13)

L∞ (Ω)

i=1

under the hypothesis (H2), (2.12) is true for Tm = ∞.

2N Proof. To show (2.12) we proceed as in [4]. Let {uN i }i=1 be the solution N to (P ). Taking in (2.11) ( 2 i for i ≥ M, (2.14) gi = 0 for 1 ≤ i < M, P2N and using (H1) and Lemma 2.2 we obtain for %N = i=M i2 uN i ,

∂%N − d∆%N ≤ A ∂t

X M ≤i,j≤N

A + 2

X

N i2 j 2 uN i uj + A

N i2 j(i + 2j)uN i uj

1≤i M.

i=M

∂ N η − d∆η N ∂t

N i(i + j)uN i uj +

1≤i 0.

Ω j=1

Similarly, by (H3) and (2.3) we have Z X N (3.7) sup bi,j−i uN ci γi M0 j (x, t) dx ≤ e t∈[0,T ]

for any T > 0.

Ω j=i+1

Let us denote the reaction terms in the ith equation of (PN ) by N N FiN = fiN − uN i gi + hi ,


289

where fiN

1 = 2

giN =

X i−1

N ai−j,j uN j−i uj

bi−j,j uN i

−

,

j=1

N X

aij uN j ,

hN i =

j=1

N −i X

bij uN i+j =

j=1

N X

bi,j−i uN j

j=i+1

if i ≤ N , and fiN =

N 1 X N aj,i−j uN j ui−j , 2

giN = 0,

hN i =0

j=i−N

for N < i ≤ 2N . By Lemma 2.2 and (3.6), (3.7) it follows that for N ≥ i, (3.8)

N N kfiN − uN i gi + hi kL∞ (0,T ;L1 (Ω)) ≤ Di ,

where Di , i = 1, 2, . . . , are positive constants independent of N . The operator L is closable in L1 (Ω) and accretive in this space. Its closure L1 generates a compact, positive, analytic semigroup in L1 (Ω) (see [1]). Using (3.8) and applying to each equation the compactness result from [3, The∞ orem 1(i)] yields, for i = 1, 2, . . . , relative compactness of {uN i }N =i in the space 1 ∞ C([0, T ]; L (Ω)). Let {Nl }l=1 be the diagonal increasing sequence such that (3.9)

l uN i → ui

in C([0, T ]; L1 (Ω)),

l uN i → ui

a.e. in ΩT

l as l → ∞, for all i = 1, 2, . . . For fixed l and i < Nl the function uN i is the mild solution given by the Duhamel formula Z t Nl Nl tdi L l l (3.10) uN (t) = e U + e(t−s)di L (fiNl (s) − uN i i i (s)gi (s) + hi (s)) ds.

0

Notice that thanks to (3.6) and (3.7), for fixed i, (3.4) follows. By (3.9) and Lemma 2.2 it follows that i−1 1 X ai−j,j ui−j uj − bi−j,j ui in C([0, T ]; L1 (Ω)) fiNl → 2 j=1 as l → ∞. In order to pass to the limit in (3.10) we shall show that for each i = 1, 2, . . . , (3.11)

(3.12)

giNl → l hN i →

∞ X

aij uj

j=1 ∞ X j=i+1

bi,j−i uj

in L1 (ΩT ), in L1 (ΩT )

290

D. Wrzosek

as l → ∞. To this end notice that by (3.1) for fixed i and arbitrary ε > 0 there exists l0 such that for any l > l0 , Z Z Z Z Nl ∞ X X ε ε l < aij uN and aij uj < , j 3 3 ΩT ΩT j=Nl0

j=Nl0

where ε is such that 3M0 T aij /j ≤ ε

for j > Nl0 .

Thanks to (3.9) there exists l1 ≥ l0 such that for any l > l1 , Nl0 −1

Z Z

X

ΩT

It follows that for l > l1 , Z Z ΩT

l aij |uN j − uj | ≤

j=1

ε . 3

X ∞ X Nl Nl aij uj − aij uj < ε. j=1

j=1

Hence (3.11) follows. The proof of (3.12) is similar, so we skip it. By Lemma 2.2, (3.9) and (3.11) it follows that l Nl uN i gi → ui

∞ X

aij uj

in L1 (ΩT ) as l → ∞,

j=1

which enables us to pass to the limit in (3.10). Using the above reasoning one can construct a solution defined on Ω × (0, ∞) which satisfies (3.2), in the following way. Let {Tn }∞ n=1 be any increasing sequence of positive numbers such that Tn → ∞. Then using a compactness argument there exists a sequence {Nl1 }∞ l=1 such that for each i = 1, 2, . . . a solution N1

2 u1i to (P) on the interval [0, T1 ] is defined as the limit of {ui l |[0,T1 ] }∞ l=1 . Let Nl N2

∞ 2 l be a subsequence of {Nl1 }∞ l=1 such that {ui |[0,T2 ] }l=1 tends to the solution ui defined on [0, T2 ]. Of course, the solutions u1i and u2i coincide on [0, T1 ]. Step by step we define in this way the sequence {Nln }∞ l=1 for any n ≥ 1. Hence, the soluNl

l ∞ tion ui to (P) on Ω × (0, ∞) is defined as the limit of {ui l }∞ l=1 , where {Nl }l=1 is the diagonal subsequence. Passing to the limit we obtain (3.4) and (3.5) from (3.8) and (2.3) respectively.

Remark 1. Since the reaction terms in each equation are in the space L∞ (0, T ; L1 (Ω)), we have ui ∈ C α ([δ, T ]; L1 (Ω)) for any 0 ≤ α < 1 and δ > 0, which follows from the regularity result for mild solutions (see e.g. [12, pp. 110–111]) applied separately to each equation. Assuming some structural assumptions on the coefficients aij we are able to prove the existence of strong solutions in L2 (ΩT ).

291


Theorem 3.2. Under the assumption (H3), suppose that there exists {ri }∞ i=1 such that (3.13)

aij = ri rj ,

(3.14)

ri = o(i) for i ≥ 1, and X bij ≤ Bk where B > 0.

(3.15)

i+j=k

Then there exists a nonnegative solution {ui }∞ i=1 to (P) defined on Ω × (0, ∞) such that (3.2) and (3.5) hold and for i = 1, 2, . . . , ui ∈ C([0, T ]; L2 (Ω))

for any T and satisfies (1.2),

1,2 Wloc (]0, ∞[; L2 (Ω))

ui ∈ ∞ X ri ui ∈ L2 (ΩT )

(3.16)

∩ L2loc (]0, ∞[; D(L)),

for any T > 0,

i=1

and the equations (1.1) are satisfied a.e. on Ω × (0, ∞). 2N N Proof. Let {uN i }i=1 be the solution of (P ). Taking gi = 1 for i ≥ 1 in (2.2) and integrating on Ωt we obtain, using (3.13),

(3.17)

2N Z X i=1

uN i (x, t) dx

Ω

−

2N Z X i=1

X N

Z Z

1 Ui (x) dx + 2 Ω

Ωt

1 = 2 Hence, using (3.15) for any T > 0,

N

2

X

N

ri u i (3.18)

i=1

ri u N i

2

i=1

Z Z

N X X

bij uN k .

ΩT k=2 i+j=k

≤ BT M0 .

L2 (ΩT )

By (H3) we have for i = 1, 2, . . . , Z Z (3.19) ΩT

X N j=i+1

bi,j−i uN j

2

Z Z ≤ γi ri ΩT

X N

rj u N j

2 .

j=i+1

Therefore, (3.20)

N N kfiN − uN i gi + hi kL2 (ΩT ) ≤ Di ,

where Di , i = 1, 2, . . . , are positive constants independent of N . Now, to end the proof, it is sufficient to use the existence result of Theorem 3.1. Since now the reaction terms are in the space L2 (Ω) the regularity of solutions in (3.16) follows from the standard theory of parabolic equations (see e.g. [11, 13, 12]) applied separately to each equation in (P).

292

D. Wrzosek

4. Existence of mass preserving solutions In this section we exploit the hypothesis (H4) which enables us to find better estimates of the terms ∞ X

∞ X

iui ,

i=1

∞ X

aij uj ,

j=1

bi,j−i uj .

j=i+1

Then the hypothesis (H3) can be modified in the following way: (H3)M For each i ≥ 1, if 1 ≤ i < M then there exists γi > 0 such that bi,j−i ≤ γi aij

(4.1)

2

for j ≥ i + 1, and for each i ≥ M.

bij = o(j )

Theorem 4.1. Under the assumptions (H4), (H3)M and (H1) or (H2), if ∞ X

i2 kUi kL∞ (Ω) ≤ const

i=1

then there exists T > 0 such that (P) has a nonnegative solution {ui }∞ i=1 on ΩT such that for i = 1, 2, . . . , ui ∈ C([0, T ]; L2 (Ω)) ∞

and satisfies (1.2),

1,2 Wloc (]0, T ]; L2 (Ω)),

(4.2)

ui ∈ L (ΩT ) ∩

(4.3)

ui ∈ L2loc (]0, T ]; D(L)), ∞ ∞ X X aij uj , bi,j−i uj ∈ L∞ (ΩT ). j=1

j=i+1

The equations of (1.1) are satisfied a.e. on ΩT and ∞ Z X (4.4) kuk (x, t) dx = M0 for t ∈ [0, T ]. k=1

Ω

Under the hypothesis (H2) the solution is defined on Ω × (0, ∞) and the above statement is true for any T > 0. Proof. We proceed in much the same way as in the proof of Theorem 3.1. By Lemma 2.3 there exists Tm > 0 such that for any N > 1 and 0 < T < Tm ,

X

X

N

N

N N

(4.5) a u ≤ Ai ju ≤ AiC1 (T ) ij j j

j=1

L∞ (ΩT )

j=1

L∞ (ΩT )

and by (4.1), (4.6)

X

N

N

b u i,j−i j

j=i+1

L∞ (ΩT )

≤ cei C1 (T ),


293

where cei is a positive constant. Under the assumption (H2) the estimates above are valid for any T > 0. It follows that (3.20) still holds true. Using again the compactness argument we can choose a subsequence {Nl }∞ l=1 such that l uN i → ui

l uN i → ui

in C([0, T ]; L2 (Ω)),

a.e. in ΩT

as l → ∞. By Lemma 2.3 it follows that for any k ≤ Nl ,

2Nl

Nl

X N

X

iA + 1 Nl l

≤ + juj (·, t) (4.7) aij uj C1 (T )

k L∞ (Ω) L∞ (ΩT ) j=k

j=k

for 0 ≤ t ≤ T. By Lemma 2.3 and (4.1) it follows that for each i ≥ 1 and for ε > 0 there exists k0 > i such that

Nl

X Nl

< ε, b u i,j−i j

L∞ (ΩT )

j=k0

where

bi,j−i < ε for j ≥ k0 . j2 Proceeding as in the last part of the proof of Theorem 3.1 one shows (3.11) and (3.12) with L1 (ΩT ) replaced by L2 (ΩT ). Similarly, by (4.7) for 0 ≤ t ≤ T , C1 (T )

2Nl X

l juN j (·, t) →

j=1

∞ X

in L1 (Ω)

juj (·, t)

as l → ∞.

j=1

Hence, using (2.3) we obtain (4.4). Finally, by (4.5) and (4.6) we arrive at (4.3) by passing to the limit. The following theorem shows the existence of solutions under the assumption (H2) for arbitrarily fast fragmentation. However, to prove it we need all diffusion coefficients to be the same in each equation. Theorem 4.2. In the case (H2), if di = d > 0 for i ≥ 1 and ∞ X

(4.8)

i2 kUi kL∞ ≤ const,

i=1

then for arbitrary bij ≥ 0, i, j ≥ 1, there exists a mild solution {ui }∞ i=1 to (P) defined on Ω × (0, ∞) such that for any T > 0 and each i = 1, 2 . . . , ui ∈ C([0, T ]; L1 (Ω)) and satisfies (1.2), ∞ ∞ X X aij uj ∈ L∞ (ΩT ), bi,j−i uj ∈ L1 (ΩT ), j=1

Z X ∞ Ω i=1

j=1

iui (x, t) dx = M0

for t ∈ [0, T ].

294

D. Wrzosek

2N N Proof. For any N > 2 let {uN i }i=1 be the solution of (P ). Setting M = 1 in Lemma 2.3 we have for T > 0,

X

2N 2 N

≤ C1 (T ) j u (4.9) j

∞ L

j=1

(ΩT )

without any growth condition on bij . In particular, it follows that the solution is global in time. Now, taking in (2.2) gi = i2 for i = 1, . . . , 2N, integrating over Ωt and using (H2) we obtain for 0 < t ≤ T , (4.10)

Z X 2N

i2 uN i (x, t) dx

Z Z

X

+

Ω i=1

ijbij uN i+j

Ωt 1 i there exists a constant cbi independent of N such that for i = 1, 2, . . . , Z Z N −i X (4.11) bij uN i+j ≤ cbi (T ). ΩT j=1

For each i inequalities (4.10) and (4.11) imply the uniform bound N N kfiN − uN i gi + hi kL1 (0,T ;L1 (Ω)) ≤ const.

Using the compactness argument from [3, Theorem 1(ii)] one can choose a subsequence {Nl }∞ l=1 such that l uN i → ui

in Lq (0, T ; L1 (Ω)) for any 1 ≤ q < ∞,

l uN i → ui

a.e. in ΩT

as l → ∞, for all i = 1, 2, . . . In the same way as in the proof of Theorem 3.1 one shows (3.11) using (4.9). To show (3.12) notice that due to (4.10) for each i ≥ 1 and k < Nl − i, Z Z NX l −i cbi (T ) Nl . bij ui+j ≤ k ΩT j=k

Now, passing to the limit in the Duhamel formula as l → ∞ we obtain l uN i → ui

in C([0, T ]; L1 (Ω)).

By (4.10) for 0 < t ≤ T we have Z X 2N Ω i=k

hence (1.4) follows.

iuN i (x, t)dx ≤

const , k


295

Remark 2. In [2] the existence of solutions for the model without diffusion is proved under the same assumptions on aij and bij as in the above theorem. It is worth pointing out that the result is shown there under the only assumption that the total mass contained in the initial data is finite, M0 < ∞. We assume more, (4.8), but our proof seems to be shorter than that in [2]. It is not known to the author how to transfer the result from [2] to our case. Remark 3. In general there is no uniqueness of solutions for the system (1.1) even for the case without diffusion (see [2]). In the particular case of aij = ri rj and ri = Ai + B where A, B are constants (A > 0, B ≥ 0), bij = 0 for i, j ≥ 1, and (H4) holds, uniqueness of solutions is proved in [4]. Acknowledgments. The author would like to express his thanks to Professor Philippe Bénilan for many discussions during the author’s stay at the Université de Franche-Comté in Besan¸con in September 1995.

References [1]

H. Amann, Dual semigroups and second order linear elliptic boundary value problems, Israel J. Math. 45 (1983), 225–254.

[2]

J. M. Ball and J. Carr, The discrete coagulation-fragmentation equations: existence, uniqueness and density conservation, J. Statist. Phys. 61 (1990), 203–234.

[3]

P. Baras, Compacit´ e de l’op´ erateur f → u solution d’une ´ equation non lin´ eaire f ∈ (du/dt) + Au, C. R. Acad. Sci. Paris S´ er. A 286 (1978), 1113–1116.

[4]

P. B´ enilan and D. Wrzosek, On an infinite system of reaction-diffusion equations, Adv. Math. Sci. Appl. 7 (1997), 349–364.

[5]

J. F. Collet and F. Poupaud, Existence of solutions to coagulation-fragmentation systems with diffusion, Transport Theory Statist. Phys. 25 (1996), 503–513.

[6]

P. Deuflhard and M. Wulkow, Computational treatment of polyreaction kinetics by orthogonal polynomials of a discrete variable, Konrad-Zuse-Zentrum (Berlin), Preprint SC 88-6 (1988).

[7]

P. G. J. van Dongen, Spatial fluctuations in reaction-limited aggregation, J. Statist. Phys. 54 (1989), 221–271.

[8]

R. Drake, Topics in Current Aerosol Research (G. M. Hidy and J. R. Brock, eds.), Pergamon Press, Oxford, 1972.

[9]

E. M. Hendriks, M. H. Ernst and R. M. Ziff, Coagulation equations with gelation, J. Statist. Phys. 31 (1983), 519–563.

[10]

F. Leyvraz and H. R. Tschudi, Singularities in the kinetics of coagulation processes, J. Phys. A. Math. Gen. 14 (1981), 3389–3405.

[11]

R. H. Martin and M. Pierre, Nonlinear reaction-diffusion systems, Nonlinear Equations in Applied Science (W. F. Ames and C. Rogers, eds.), Academic Press, Boston, 1992.

[12]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.

[13]

F. Rothe, Global solutions of reaction-diffusion systems, Springer-Verlag, Berlin, 1984.

[14]

M. Slemrod, Coagulation-diffusion systems: derivation and existence of solutions for the diffuse interface structure equations, Phys. D 46 (1990), 351–366.

296 [15]

D. Wrzosek M. Smoluchowski, Versuch einer mathematischen Theorie der kolloiden L¨ osungen, Z. Phys. Chem. 92 (1917), 129–168.

[16]

R. M. Ziff, Kinetics of polymerization, J. Statist. Phys. 23 (1980), 241–263.

[17]

R. M. Ziff and G. Stell, Kinetics of polymer gelation, J. Chem. Phys. 73 (1980), 3492–3499.

Manuscript received September 20, 1996

Dariusz Wrzosek Department of Mathematics, Computer Science and Mechanics University of Warsaw Banacha 2 02-097 Warszawa, POLAND E-mail address: [email protected]

TMNA : Volume 9 – 1997 – No 2