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Convergence in a multi-layer population model with age-structure
Nonlinear Analysis: Real World Applications 8 (2007) 887 – 902 www.elsevier.com/locate/na
Convergence in a multi-layer population model with age-structure夡 Caterina Cusulina , Mimmo Iannellia,∗ , Gabriela Marinoschib a Dipartimento di Matematica, Università di Trento, 38050 Povo (Trento), Italy b Institute of Mathematical Statistics and Applied Mathematics, POB 1-24 Bucharest, Romania
Received 28 April 2005; accepted 23 March 2006
Abstract In this paper we investigate the convergence of a multi-layer population model to a single-layer limit. In a previous paper [Cusulin, C., Iannelli, M., Marinoschi, G. Age-structured diffusion in a multi-layer environment, Nonlinear Anal. Real World Appl. 6(1) (2005) 207–223], we considered a Gurtin–MacCamy type model based on the fact that the population diffuses through a one dimensional habitat, partitioned into n homogeneous layers. In each layer the population has its own age-dependent growth and diffusion parameters, so that within each layer the dynamics is not subject to environmental variations, while changes occur from a layer to another, according to different conditions. Such kind of a model may describe the growth of a population in a fragmented environment, but the same framework may be used to give an approximate view of the population growth and diffusion in a general spatially heterogeneous habitat, because the layer structure may arise by approximation of the original problem. In the present paper we show that this view is actually mathematically sound and justified. In fact, on the basis of the previous results (see [Cusulin, C., Iannelli, M., Marinoschi, G. Age-structured diffusion in a multi-layer environment, Nonlinear Anal. Real World Appl. 6(1) (2005) 207–223]) the approximating problem actually converges and the multi-layer solution may be considered a patch-wise picture of the original problem. 䉷 2006 Elsevier Ltd. All rights reserved. Keywords: Evolution equations; Reaction diffusion; Multi-layer; m-accretive operators; Age structure; Population dynamics
1. Brief introduction and statement of the problem In a previous paper [6] we have considered the problem of an age-structured population diffusing in a one dimensional environment partitioned into n different layers. In each layer the population has its own growth and diffusion parameters, so that within each layer the dynamics is not subject to environmental variations, while changes occur from a layer to another, according to the different conditions. Such kind of a model may describe the growth of a population in a heterogeneous habitat, fragmented into patches where the diffusion coefficient and the vital rates are constant with respect to the spatial variable, and vary only from one patch to another. Our attention here is focused on the one dimensional case, but the approach could be extended to higher dimensions. 夡 This research was supported in part within the FIRB project RBAU01K7M2 “Metodi dell’Analisi Matematica in Biologia, Medicina e Ambiente” of the Italian Ministero Istruzione Università e Ricerca and GAR 22, 2005-2006 of the Romanian Academy. ∗ Corresponding author. Tel.: +39 461 881657; fax: +39 461 881624. E-mail addresses: [email protected] (C. Cusulin), [email protected] (M. Iannelli), [email protected] (G. Marinoschi).
1468-1218/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2006.03.012
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Models for population diffusion in a fragmented habitat have been recently considered by many authors. In particular, [2,3,8,10,11] are concerned with an alternance of patches with different vital rates. In the formulation considered in [6] we have joined diffusion and age structure, proving well-posedness of the problem. Actually, we may note that the framework of the multi-layer model may arise as a stepwise approximation of a model formulated with general diffusion coefficient and vital rates as functions of the space variable. In fact, such functions may be approximated stepwise and a sequence of multi-layer problems is generated as an approximation of the original problem. In the present paper we give a precise formulation of this view and show that it is actually mathematically sound and justified. In fact, on the basis of the previous results (see [6]) we prove that an approximating sequence of problems actually converges and that the multi-layer solution may be considered a patch-wise picture of the original problem. To set a precise formulation of the problem, we denote = (0, a + ) × (y0 , yL ), 0 = {(0, y); y ∈ (y0 , yL )},
a + = {(a + , y); y ∈ (y0 , yL )},
y0 = {(a, y0 ); a ∈ (0, a + )},
yL = {(a, yL ); a ∈ (0, a + )}
and consider the following problem in : jp jp j + + (a, y, S(t, y))p − jt ja jy
a+
p(t, 0, y) =
K(a, y)
(a, y, S(t, y))p(a, t, y) da,
jp jy
=f
in (0, T ) × ,
(1)
(2)
0
S(t, y) =
(a, y, z)p(t, a, z) dz da,
(3)
with the boundary conditions p(0, a, y) = p0 (a, y)
in ,
(4)
K(a, y)
jp =0 jy
on (0, T ) × y0 ,
(5)
K(a, y)
jp =0 jy
on (0, T ) × yL .
(6)
This will be called problem (P ). In the model above a ∈ (0, a + ) is the age, y ∈ (y0 , yL ) is the spatial variable and t ∈ (0, T ) is the time. Moreover, K(a, y), (a, y, x), (a, y, x), are, respectively, the diffusion coefficient, mortality and fertility. Finally, f is some source in the layer and (a, y, z) is a weight function that through the size (3) models how the values of the solution at different points of the environment influence the population at the point y. We note that the problem is actually autonomous, because the way the size S(t, y) depends upon t is actually induced by the solution (see (3)). Besides this, in order to introduce the n-layer structure we denote j = (0, a + ) × (yj −1 , yj ), yj = {(a, yj ); a ∈ (0, a + )},
j = 1, . . . , n, j = 0, . . . , n
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and consider the following problem for j = 1, 2, . . . , n, on the variables pj (t, a, y) defined on [0, T ] × j ; jpjn jt
jpjn
+
ja
+ nj (a, Sjn (t))pjn − Kjn (a)
j2 pjn jy 2
= fjn
in (0, T ) × j ,
(7)
n pjn (0, a, y) = p0,j (a, y) in j ,
pjn (t, 0, y) = Sjn (t) =
a+ 0
n
a+
k=1
nj (a, Sjn (t))pjn (a, t, y) da,
yk yk−1
0
(8) (9)
nj (a, z)pkn (t, a, z) dz da,
(10)
endowed with the boundary conditions K1n (a)
jp1n =0 jy
on (0, T ) × y0 ,
(11)
Knn (a)
jpnn =0 jy
on (0, T ) × yL ,
(12)
and with the additional conditions on each interface between two layers (for j = 1, 2, . . . , n − 1) pjn = pjn+1 Kjn (a)
jpjn jy
on (0, T ) × yj , = Kjn+1 (a)
jpjn+1 jy
(13) on (0, T ) × yj .
(14)
Here we are using the superscript “n”, since this problem is associated to the partition of the space interval (y0 , yL ) in n layers; we shall refer further to it as problem (P n ). All the functions Kjn , nj , nj , fjn , nj are approximations of the respective functions in problem (P ) and all of them, but nj , are constant within each layer, with respect to the space variable y. Precise assumptions about all these functions and on the way they converge to their respective limits will be stated later within the functional framework that will be set up in the next section. 2. Functional framework The goal of this section is the formulation of a functional framework to treat problems (P ) and (P n ), following the procedure used in [6]. Namely, we want to formulate these problems as abstract Cauchy problems in the space H = L2 (). To this aim we consider the spaces (see [6]) V = H 1 (y0 , yL ),
H = L2 (y0 , yL ),
and, if V is the dual of V , we have V ⊂ H ⊂ V . Then, we consider problem (P ) and define first the operator A0 : D(A0 ) ⊂ L2 (0, a + ; V ) → L2 (0, a + ; V ), on the domain
+
+
a+
D(A0 ) = u ∈ L (0, a ; V ); ua ∈ L (0, a ; V ), u(0, y) = 2
2
0
(a, y, S(y))u(a, y) da ,
(15)
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by the relation
A0 u, = ua , +
((a, y, S(y))u + K(a, y)uy y ) da dy,
(16)
∀ ∈ L2 (0, a + ; V ). By ·, · we denote the duality product between the space L2 (0, a + ; V ) and its dual L2 (0, a + ; V ); we note that a+ f, g = f (a), g(a)V ,V da. 0
We specify that in (16) we set (a, y, z)u(a, z) dz da. S(y) =
(17)
Actually, this is an operator S : H → H defined by (a, y, z)u(a, z) dz da, (Su)(y) =
but for the writing simplicity we skip this notation and use (17). Also we notice that u at a = 0 makes sense, since u ∈ L2 (0, a + ; V ) and ua ∈ L2 (0, a + ; V ) implies u ∈ C([0, a + ]; H ). Finally, we define the operator A : D(A) ⊂ H → H by setting D(A) = {u ∈ D(A0 ), A0 u ∈ H },
Au = A0 u,
∀u ∈ D(A),
and we are led to the Cauchy problem corresponding to (P ) dp + Ap = f dt p(0) = p0 .
a.e. t ∈ (0, T ),
(18) (19)
To place problems (P n ) within the same framework we need first to build new functions on (0, T ) × by a stepwise definition n p1 (t, a, y), y ∈ (y0 , y1 ), n (20) p (t, a, y) = · · · pnn (t, a, y), y ∈ (yn−1 , yL ), n S1 (t), y ∈ (y0 , y1 ), n (21) S (t, y) = · · · Snn (t), y ∈ (yn−1 , yL ), n 1 (a, x), y ∈ (y0 , y1 ), (22) n (a, y, x) = · · · nn (a, x), y ∈ (yn−1 , yL ), n 1 (a, x), y ∈ (y0 , y1 ), n (23) (a, y, x) = · · · nn (a, x), y ∈ (yn−1 , yL ), n 1 (a, z), y ∈ (y0 , y1 ), z ∈ (y0 , yL ), n (24) (a, y, z) = · · · nn (a, z), y ∈ (yn−1 , yL ), z ∈ (y0 , yL ),
C. Cusulin et al. / Nonlinear Analysis: Real World Applications 8 (2007) 887 – 902
n
K (a, y) = p0n (a, y) =
K1n (a), ··· Knn (a),
y ∈ (y0 , y1 ), (25) y ∈ (yn−1 , yL ),
n (a, y), p0,1 ··· n (a, y), p0,n
n
f (a, y) =
891
f1n (a, y), ··· fnn (a, y),
y ∈ (y0 , y1 ), (26) y ∈ (yn−1 , yL ), y ∈ (y0 , y1 ), (27) y ∈ (yn−1 , yL ).
We note that after these definitions S n (t, y) can be rewritten (see (10)) as S n (t, y) = n (a, y, z)p(t, a, z) dz da.
(28)
At this point we stress again that S n (y) is in fact the operator S n : H → H defined by n (S u)(y) = n (a, y, z)u(a, z) da dz.
(29)
Now we may proceed as in the previous definition of the operator A in order to define the following sequence of operators An : D(An ) ⊂ H → H by setting An u = An0 u, on the domain D(An ) = {u ∈ D(An0 ),
An0 u ∈ H },
where An0 : D(An0 ) ⊂ L2 (0, a + ; V ) → L2 (0, a + ; V ) is defined as An0 u, = ua , +
(n (a, y, S n (y))u + K n (a, y)uy y ) da dy,
for all ∈ L2 (0, a + ; V ), on the domain D(An0 ) = u ∈ L2 (0, a + ; V ); ua ∈ L2 (0, a + ; V ), u(0, y) =
a+
n (a, y, S n (y))u(a, y) da .
0
By all these we are led to the Cauchy problem corresponding to (P n ) dpn + An p n = f n dt pn (0) = p0n .
a.e. t ∈ (0, T ),
(30) (31)
Therefore, we can notice that problems (P ) and (P n ) are the same as mathematical form, but the equations have different coefficients. For each problem (P n ) we have existence and uniqueness results obtained in [6] under certain hypotheses and consequently we have similar results for problem (P ) with the corresponding replacements of the functions involved in it. The next section is devoted to a review of the results obtained in [6] and to the introduction of the assumption on our new problem.
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3. Problem hypotheses and review of the existence results Here we summarize the assumptions and briefly explain the way in which the solution to (30)–(31) or (18)–(19) was constructed in [6]: this will be a basis to prove further results about convergence. Actually our assumptions on problems (P n ) repeat those in [6] but also introduce uniformity with respect to the index n. Namely we assume that, for each n, all the functions n (a, y, x), n (a, y, x), n (a, y, z) and K n (a, y) are measurable and we suppose that, for each R > 0, and any x, x ∈ R with |x| R, |x|R, there exist L (R) > 0 and L (R) > 0, such that |n (a, y, x) − n (a, y, x)| L (R)|x − x|,
(32)
|n (a, y, x) − n (a, y, x)| L (R)|x − x|,
(33)
uniformly with respect to a and y. Moreover, we assume that we have 0 n (a, y, x)+
(34)
0 n (a, y, x) with n (a, y, 0) = 0,
(35)
0 n (a, y, z)∞ ,
(36)
0 < K0 K n (a, y) K∞ .
(37)
and
Note that our assumptions directly concern the functions defined on and built in (20)–(27). Of course, such properties may be drawn from those of the components in each layer as done in [6]. Besides this set of assumptions, which includes the basic properties of the functions involved in problem (P n ), we shall impose also the following set of convergence hypotheses to the functions involved in problem (P ). n→∞
n (a, y, x) → (a, y, x) uniformly with respect to a, y and x, n→∞
n (a, y, x) → (a, y, x) uniformly with respect to a, y and x, n→∞
n (a, y, z) → (a, y, z) uniformly with respect to a, y and z, n→∞
K n (a, y) → K(a, y) uniformly with respect to a and y.
(38) (39) (40) (41)
We note that such assumptions (38)–(41) also imply that the same properties (32)–(37) are satisfied by the functions (a, y, x), (a, y, x), (a, y, z), K(a, y) involved in problem (P ) (with the same constants L (R), L (R), + , ∞ , K∞ and K0 ). Thus we have that all the results in [6] can be applied both to (P n ) and (P ). In fact, such results provide the following statement: Theorem 1. Under assumptions (32)–(37), if f n ∈ W 1,1 (0, T ; H ), p0n ∈ D(An ), problem (P n ) has a unique solution pn ∈ W 1,∞ (0, T ; H ) ∩ L∞ (0, T ; D(An )) and it satisfies the estimate T 1 p n (t) 2H p0n 2H + f n (t) 2H dt exp[(2+ a + + 2)t], (42) Kmin 0 with Kmin = min(1, K0 ). The same result holds also for problem (P ). To give an idea of the proof, we will give a few elements which will be useful for further developments. For writing simplicity, in the next explanations we shall refer to the notations for problem (P ).
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The basic step of the proof was to reduce the problem to be solved under the assumptions (32), (33) to another one in which these two assumptions are replaced by global Lipschitz conditions on H for the functions E(u) : H → H and F (u) : H → H defined as follows: E(u) ≡ (a, y, S(y))u(a, y), F (u) ≡ (a, y, S(y))u(a, y), (43) where S(y) = (a, y, z)u(a, y) da dz. In fact, these functions are only locally Lipschitz-continuous and, for our purposes, they were truncated by introducing the approximated functions E(u) for u H N, Nu EN (u) = (44) E for u H > N u H and
FN (u) =
F (u) F
Nu u H
for u H N, for u H > N,
(45)
which are Lipschitz continuous on H , for each N fixed. With these definitions, we consider for each N the corresponding operator AN that coincides with A on the ball of radius N. AN can be proved to be an m-accretive operator so that we have an existence result for the Cauchy dpN + A N pN = f dt
a.e. t ∈ (0, T ),
(46)
pN (0) = p0 ,
(47)
and, since we also can prove the following estimate (see (42)): T 1 2 2 2 pN (t) H N+ = p0 H + f (t) H dt exp[(2+ a + + 2)T ], Kmin 0
(48)
we find that, taking N 2 > N+ , the solution to (46)–(47) is actually a solution of (18)–(19). Moreover, another estimate can be deduced for this problem, that is, T 2 2 2 (f − f )(t) H dt e0 (N)t , (49) p(t) − p(t) H p0 − p0 H + 0
with 0 (N ) = B 2 (N )a + + 2M(N) + 1, where p and p are two solutions corresponding to p0 , f on the one hand and p0 , f on the other hand. Here M(N ) and B(N) are the global Lipschitz constants for EN and FN on H , i.e.,
where
and
EN (u) − EN (u) H M(N ) u − u H ,
(50)
FN (u) − FN (u) H B(N) u − u H ,
(51)
√ √ M(N ) = 2∞ a + yL − y0 L (N )N √ √ B(N ) = ∞ a + yL − y0 L (N )N + + .
In view of the previous sketch of the proof, which holds for the problems (P n ) as well, we conclude that it is sufficient to develop the theory for the truncated problems (because, as we have seen, the solutions to the original problems are in fact the solutions to the truncated ones). n (t) to p (t) and with the operators We mention now that furthermore we shall deal with the truncated solutions pN N n AN and AN associated to them. Since the notation has become too cumbersome, we shall omit the subscript N ; however, the original functions coincide with the truncated ones in the ball of radius N , N being fixed by (48).
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In order to prove the convergence of the solution to (P n ) to the solution to (P ) we will use the following result, given in [4], concerning a sequence of quasi m-accretive operators in a Banach space X, An : D(An ) ⊂ X → X and their “limit” A : D(A) ⊂ X → X, stated in terms of their resolvents. Theorem 2. Let An and A be quasi m-accretive operators and let Un (t) and U(t) be the semigroups generated by −An and −A, respectively. If lim ( + An )−1 g = ( + A)−1 g
n→∞
for every g ∈ D and > 0 , where D =
n 1 D(A
n
) ∩ D(A) and 0 is independent of n, then
lim Un (t)g = U(t)g
n→∞
for every g ∈ D and the limit is uniform on bounded intervals for t. Thus, as seen in the previous theorem, the central point of the study remains the proof of the convergence of the sequence of resolvents. To come to this end, certain intermediate results, including the determination of the closure of the domain of definitions of the operators An and A, some convergence results and estimate settlements for the solutions to the resolvent problems are necessary. The next section is devoted to these preliminaries. 4. Study of the resolvents of the operators An and A Here we consider the m-accretive operators A and An defined in the previous sections under assumptions (32)–(37). We recall that these operators are indeed the truncations AN and AnN defined through the functions FN (·), FNn (·) and n (·), introduced in (44)–(45). We start the study by a density result. EN (·), EN Proposition 1. The domains of the operators A and An satisfy D(A) = D(An ) = H . Proof. Of course, we consider only the operator A, the proof for An being identical. First, let us consider the problem du + A (a)u = 0, da u(0) = u0 ,
(52)
where the linear operator A (a) : V → V is defined by yL A (a)u, V ,V = K(a, y)uy y dy, ∀u ∈ V , ∈ V . y0
(53)
It is known that, if u0 ∈ H , this problem has a solution that we denote by u(a, y; u0 ), such that u ∈ L2 (0, a + ; V ),
ua ∈ L2 (0, a + ; V ),
u(a, ·; u0 ) H u0 H .
(54) (55)
Let now f ∈ H and consider a sequence fn ∈ C ∞ ([0, a + ) × [y0 , yL ]) such that fn → f in H . Let us introduce the mapping Tn : L2 (0, a + ; V ) → H , (Tn g)(a, y) = (na)fn (a, y) + (1 − (na))u(a, y; g), where the mapping : H → H is defined by a+ ( g)(y) = F (g)( , y) d 0
(56)
(57)
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and ∈ C ∞ (R+ ) is a function such that (see [1]) 0 (x)1, (x) = 0
for 0 x 1/2,
(x) = 1 for x 1.
(58)
We note that, according to the definition of F in (43) and to (51), the mapping satisfies √ g − h H B(N) a + g − h H
(59)
and that Tn g ∈ L2 (0, a + ; V ),
(Tn g)a ∈ L2 (0, a + ; V ).
(60)
Moreover, for g, h ∈ H we have yL 1/n Tn g − Tn h 2H = |(1 − (na))u(a, y; g − h)|2 da dy
y0 0 1/n yL 0
y0
|u(a, y; g − h)|2 dy da
B(N)2 a + 1 g − h 2H . g − h 2H n n Here we took into account that for na 1 we have 1 − (na) = 0 and for na 1 we have 1 − (na) 1. Consequently, the integral on [1/n, a + ] vanishes. We used also (55) and (59). In fact, we obtained that Tn is a contraction on H and then the equation Tn g = g has a solution
+ a
gn (a, y) = (na)fn (a, y) + (1 − (na))u a, y;
F (gn ) da .
(61)
0
a+ It can be seen that gn (0, y) = 0 (a, y, S(y))gn (a, y) da and A0 gn ∈ H , so that gn ∈ D(A). Furthermore, from (61) we have 1/n yL 1 |u(a, y; gn )|2 dy da gn 2H fn 2H + 2 y0 0 1 fn 2H + gn 2H n B 2 (N )a + gn 2H , fn 2H + n so that, for n sufficiently large the sequence gn is bounded gn 2H 4 f 2H .
(62)
On the other hand, f (a, y) − gn (a, y) = (na)(f (a, y) − fn (a, y)) + (1 − (na))[f (a, y) − u(a, y; gn )], so that f − gn H f − fn H + and consequently, using (62), f = lim gn n→∞
in H ,
thus proving that D(A) = H .
1/n yL 0
y0
1/2 |f (a, y)| dy da 2
√ B(N ) a + + gn H √ n
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We notice that we have also the following additional result Proposition 2. Let f ∈ L1 (0, T ; H ),
p0 ∈ D(A) = H ,
(63)
then there exists a weak (mild) solution to the Cauchy problem (46)–(47) satisfying t 1 1 p(t) − x 2H p(s) − x 2H − (f − Ax, p() − x) d, 2 2 s ∀x ∈ H and ∀s, 0 s t T . In fact, the operator A, being m-accretive, generates a non-linear semigroup of quasi contractions on D(A) = H . We also recall that in [6] we considered the resolvent problem u + Au = g, where g ∈ D(A) = H , which can be explicited as du + AV (a)u = g a.e. a ∈ (0, a + ), da a+ (a, y, S(y))u(a, y) da. u(0) = u +
(64) (65)
0
Here the operator AV (a) : V → V is defined by yL (K(a, y)uy y + (a, y, S(y))u ) dy, AV (a)u, V ,V = y0
∀ ∈ V , so that problem (64)–(65) may be equivalently written as ua , + ( u + K(a, y)uy y + (a, y, S(y))u − g) da dy = 0,
(66)
∀ ∈ L2 (0, a + ; V ). In [6] it was shown that problem (64)–(65) has a unique solution u ∈ L2 (0, a + ; V ) with ua ∈ L2 (0, a + ; V ), for B 2 (N )a + + (B 2 (N )a + )2 + 4M 2 (N ) . > c = 2 We now need some estimates that we prove in the following: Proposition 3. The solution to the resolvent problem (64)–(65) associated to problem (P ) satisfies the estimates u 2H + u(a + , ·) 2H + K0 u 2L2 (0,a + ;V ) g 2H ,
(67)
ua L2 (0,a + ;V )
(68)
+ M(N) K∞ +√ + 1 g H , √ K0 0
[2 − (1 + B(N)2 a + + M(N ) + K0 )] u − u 2H + u(a + , ·) − u(a + , ·) 2H + K0 u − u 2L2 (0,a + ;V ) g − g 2H , where u and u are two solutions corresponding to g and g and 0 ≡ max(1 + 2+ a + + K0 , c ).
(69)
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Proof. Let g ∈ H . In order to prove estimate (67) we multiply Eq. (64) by u and integrate over . We have 1 yL 2 + 1 yL 2 u (a , y) dy − u (0, y) dy + K(a, y)u2y da dy g H u H . u 2H + 2 y0 2 y0 But
yL
y0
u2 (0, y) dy 2+ a + u 2H ,
(70)
whence we get [2 − (1 + 2+ a + + K0 )] u 2H + u(a + , ·) 2H + K0 u 2L2 (0,a + ;V ) g 2H . This implies that for > 0 we obtain (67) as claimed. Concerning (69), we take g, g ∈ H , multiply the equation (u − u) + (Au − Au) = g − g by u − u and we integrate over . Since yL (u − u)2 (0, y) dy B 2 (N )a + u − u 2H , y0
(71)
after similar calculations as before we get (69) as claimed. Finally, for estimating the norm of ua we use the definition ua L2 (0,a + ;V ) =
sup ∈L2 (0,a + ;V ), 1
|ua ()|
(72)
and calculate ua () from (66). We have
+
a
|ua ()| = ua (a), (a)V ,V da
0
=
(− u − Kuy y − (a, y, S(y))u + g) da dy
( u H + K∞ u L2 (0,a + ;V ) + M(N ) u H + g H ) L2 (0,a + ;V ) K∞ + M(N ) +√ + 1 g H L2 (0,a + ;V ) √ K0 0 and we get (68).
Before concluding this section we note that the results of Propositions 2 and 3 hold also for the sequence of operators An and all the estimates are uniform with respect to n. In particular, we have the following: Proposition 4. For each n, the sequence un = ( + An )−1 ∈ D(An ) satisfies the estimates un 2H + un (a + , ·) 2H + K0 un 2L2 (0,a + ;V ) g 2H , 2 − (1 + B 2 (N )a + + M(N ) + K0 ) un − un 2H + un (a + , ·) − un (a + , ·) 2H + K0 un − un 2L2 (0,a + ;V ) g − g 2H , and una L2 (0,a + ;V )
+ M(R) K∞ +√ + 1 g H , √ K0 0
for > 0 , with the same a + , + , ∞ , K0 , K∞ , B(N ), M(N ) and 0 for all n.
(73)
(74)
(75)
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5. Convergence of the semigroup In this section we will use Theorem 2 to prove the convergence of the sequence of the semigroups generated by −An to the semigroup generated by −A. Namely we will prove convergence for the Cauchy problems with null terms f and f n . We give first an intermediate convergence result. Lemma 1. Let (34)–(37) and (38)–(41) and assume that un → u strongly in H and weakly in L2 (0, a + ; V ). Then F n (un ) → F (u) strongly in H ,
(76)
E n (un ) → E(u) strongly in H ,
(77)
K n uny → Kuy
(78)
weakly in H .
Proof. For proving (76) we first note that, for any fixed u ∈ H and > 0, for n sufficiently large we have |F n (u)(a, y) − F (u)(a, y)| |n (a, y, S n (y)) − (a, y, S n (y))||u(a, y)| + |(a, y, S n (y)) − (a, y, S(y))||u(a, y)| |u(a, y)| + L (N )|S n (y) − S(y)||u(a, y)| |u(a, y)|(1 + L (N )), where we have used (38) and (40). Thus we have lim F n (u) = F (u),
n→∞
∀u ∈ H ,
and (76) is proved because F n (un ) − F (u) H F n (un ) − F n (u) H + F n (u) − F (u) H B(N) un − u H + F n (u) − F (u) H . Of course, (77) can be proved in the same way. Concerning (78) we note that, for any ∈ H and > 0, for n sufficiently large we have
(K n un − Kuy ) da dy y
n n n
|K − K||uy ||| da dy + K(uy − uy ) da dy
n n
ε uy H H + K(uy − uy ) da dy
.
Thus, since
uny H
is bounded and the function K(a, y)(a, y) belongs to H , we get (78).
Now, in view of Theorem 2 we prove covergence of the resolvents: Theorem 3. Let g ∈ H and assume all properties (34)–(37) and the convergence hypotheses (38)–(41). Then lim ( + An )−1 g = ( + A)−1 g
n→∞
in H .
(79)
Proof. Let us fix g ∈ H and denote un = ( + An )−1 g. We have to prove that un has a limit and this is exactly ( + A)−1 g. Assumptions (34)–(37) imply immediately (73)–(75), as specified in Proposition 4. Therefore we deduce that the sequence un is bounded in H ∩ L2 (0, a + ; V ), una is bounded in L2 (0, T ; V ) and un (a + ) lies in a bounded subset of H .
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Thus, extracting a subsequence we have the following convergence results: un → u weakly in L2 (0, a + ; V ),
(80)
una → ua
(81)
weakly in L2 (0, a + ; V ),
which should be understood in the sense of distributions and un (a + ) →
weakly in V .
(82)
From the first two relationships and since V is compact in H , we get (see [9]) that un → u
strongly in L2 (0, a + ; H ) = H .
(83)
Now, we may pass to the limit in the equality una , + ( un + K n (a, y)uny y + n (a, y, S n (y))un − g) da dy = 0,
∀ ∈ L2 (0, a + ; V ) and obtain, using (80), (81) and Lemma 1, that ua , + ( u + K(a, y)uy y + (a, y, S(y))u − g) da dy = 0,
∀ ∈ L2 (0, a + ; V ). Now it is obvious that u is the solution to (66), which is the resolvent problem associated to (P ). We can also prove that un (0) → u(0)
strongly in H
and un (a + ) → u(a + ) weakly in V . Indeed, we have
un (0) − u(0) 2H =
yL y0
+
2
a
(F n (un ) − F (u)) da dy
0
+
a F n (un ) − F (u) 2H a + B 2 (N ) un − u 2H . Then we can write un (a + , y) = un (0) +
a+ 0
una (a, y) da
and since we have just proved that un (0) → u(0) strongly in H it remains to show only that a+ 0 ua (a, y) da weakly in V . The latter follows due to (81), since we have for any ∈ V that
yL
y0
a+
(y) 0
una (a, y) da dy
yL
y0 yL
=
→ since ∈ L2 (0, a + ; V ).
y0
a+ 0 a+ 0
a+ 0
una (a, y) da →
(y)una (a, y) da dy (y)ua (a, y) da dy =
yL y0
a+
(y) 0
ua (a, y) da dy,
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6. Convergence for the Cauchy problems We now move on to the proof of the final results concerning problems (P n ) and (P ). First we take f ∈ L1 (0, T ; H ) and p0 ∈ H being the same as in problem (P ) and consider the problem (P n ) in which the free term and initial data are exactly f and p0 . d pn n = f , + An p dt n (0) = p0 . p
(84)
n ). Then we have: We shall call it (P Theorem 4. Let N be fixed, f ∈ L1 (0, T ; H ), p0 ∈ H and assume (34)–(37) and (38)–(41). Then the solution to n ) tends to the solution to problem (P ), i.e. problem (P n (t) → p(t) p
uniformly for t ∈ [0, T ].
(85)
Proof. Assume first that f ∈ W 1,1 (0, ∞; H ) and according to [7], define the following operator in the space X = H × L1 (0, ∞; H ): An : D(An ) = D(A) × W 1,1 (0, ∞; H ) ⊂ X → X by An (u, ) =
An u − (0) , −
(86)
for all (u, ) ∈ D(A) × W 1,1 (0, ∞; H ). If we denote n (t) p n , P (t) = (t)
(87)
where ∈ W 1,∞ (0, T ; W 1,1 (0, ∞; H )) is defined by (t)(s) = f (t + s),
∀s ∈ (0, ∞),
(88)
it follows that the problem dPn (t) + An Pn (t) = 0 dt p0 Pn (0) = f (s)
a.e. t ∈ (0, T ), (89)
n ). is equivalent to (P Indeed, the second component of (89) gives the equations j(t, s) j(t, s) − = 0, jt js
(0, s) = f (s),
which are necessarily satisfied by the solution (t, s) = f (t + s). Writing the first component in (89) we get d pn n (t) − (t)(0) = 0 (t) + An p dt
a.e. t ∈ (0, T ),
n ). and, since (t)(0) = f (t), we obtain the equation and the initial condition for problem (P
C. Cusulin et al. / Nonlinear Analysis: Real World Applications 8 (2007) 887 – 902
901
Of course, the same construction performed above can be repeated for problem (P ) and in the same space X we can define an operator Au − (0) , A(u, ) = − on the domain D(A) = D(A) × W 1,1 (0, ∞; H ). Further, according to the results given in [5,7] it turns out that the operators An and A are quasi m-accretive on X. Also, by Proposition 1 we can deduce immediately that D(An ) = D(A) = X. Finally, we have that the resolvent of the operator An tends to the resolvent of the operator A. To show this we fix ( g ) ∈ X and, since the operator An is quasi m-accretive, for sufficiently large we may consider n
( un ) ∈ D(An ) × W 1,1 (0, ∞; H ), given by n u n −1 g . = ( I + A ) n
This means that we have un + An un − n (0) = g, n − (n ) = . The solution to the second equation is s n s − (s) = e C− e () d
for > 0
0
and the constant C should be determined such that n ∈ W 1,1 (0, ∞; H ). Hence we impose that lims→∞ n (s) = 0 and it follows that ∞ e− () d. n (s) = e s s
We notice that this solution does not depend on n, i.e., n (s) = (s) and introducing it in the first equation we obtain ∞ n n n e− () d. u + A u = g + Since
∞ 0
0
e− () d
un n
=
∈ H we deduce by Theorem 3 that
∞ ∞ −1 g + 0 e− () d ( I + An )−1 g + 0 e− () d n→∞ ( I + A) u → = . ∞ − ∞ − s s e s e () d e s e () d
It is obvious that u g = ( + A)−1 . To conclude, resuming Theorem 2 we get then that the semigroup generated by the operator −An corresponding to n ) tends to the semigroup generated by −A, i.e. problem (P n (t) n→∞ p(t) p Pn (t) = → = P(t), (t) (t) uniformly for t ∈ [0, T ].
Now we take f ∈ L1 (0, T ; H ), extend it by 0 to (T , ∞) and consider a sequence fm ∈ W 1,1 (0, ∞; H ), such m→∞ that fm → f in L2 (0, ∞; H ). It is obvious that the result of Theorem 4 is preserved, on the basis of estimate (49) (written for problem (P n )) and it is also true for any f ∈ L2 (0, T ; H ).
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The last theorem gathers all the results obtained up to now and reaches the goal of our paper. Theorem 5. Let N be fixed, f n ∈ L2 (0, T ; H ), p0 ∈ H and assume (34)–(37), (38)–(41) and the convergence relationships fn → f
strongly in L2 (0, T ; H ),
(90)
p0n → p0
strongly in L2 (0, T ; H ).
(91)
Then lim p n (t) = p(t),
n→∞
uniformly for any t ∈ [0, T ],
(92)
where p n (t) and p(t) are the solutions to problems (P n ) and (P ), respectively. Proof. Consider the problem (P n ) corresponding to two sets of initial data and free terms, respectively, to f, p0 and n ). f n , p0n . The first one is in fact problem (P We extend f n to (0, ∞) by setting f n (t) = 0 for t ∈ [T , ∞). By (49), (90) and (91) we get that n (t) 2H → 0 pn (t) − p
as n → ∞. n→∞
n (t) → p(t), so finally we obtain that the solution p n (t) to problem (P n ) tends to the By Theorem 4 we have that p solution to problem (P ) uniformly for t ∈ [0, T ]. References [1] V. Barbu, M. Iannelli, The semigroup approach to non-linear age-structured equations, Rend. Istit. Mat. Univ. Trieste (Suppl. vol. XXVIII) (1997) 59–71. [2] H. Berestycki, F. Hamel, L. Roques, Analysis of the periodically fragmented environment model: I—species persistence, J. Math. Biol. 51 (2005) 75–113. [3] H. Berestycki, F. Hamel, L. Roques, Analysis of the periodically fragmented environment model: II—Biological invasions and pulsating travelling fronts, J. Math. Pures Appl. 84 (2005) 1101–1146. [4] H. Brezis, A. Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces, J. Funct. Anal. 9 (1972) 63–74. [5] M.G. Crandall, A. Pazy, An approximation of integrable functions by step functions with an application, Proc. Am. Math. Soc. 76 (1) (1979) 74–80. [6] C. Cusulin, M. Iannelli, G. Marinoschi, Age-structured diffusion in a multi-layer environment, Nonlinear Anal. Real World Appl. 6 (1) (2005) 207–223. [7] C.M. Dafermos, M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups, J. Functional Anal. 13 (1973) 97–106. [8] N. Kinezaki, K. Kawasaki, F. Takasu, N. Shigesada, Modeling biological invasion into periodically fragmented environments, Theor. Popul. Biol. 64 (2003) 291–302. [9] J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969. [10] N. Shigesada, K. Kawasaki, Biological invasions: theory and practice. Oxford Series in Ecology and Evolution, Oxford University Press, Oxford, 1997. [11] N. Shigesada, K. Kawasaki, E. Teramoto, Traveling periodic waves in heterogeneous environments, Theor. Popul. Biol. 30 (1986) 143–160.
Convergence in a multi-layer population model with age-structure夡 Caterina Cusulina , Mimmo Iannellia,∗ , Gabriela Marinoschib a Dipartimento di Matematica, Università di Trento, 38050 Povo (Trento), Italy b Institute of Mathematical Statistics and Applied Mathematics, POB 1-24 Bucharest, Romania
Received 28 April 2005; accepted 23 March 2006
Abstract In this paper we investigate the convergence of a multi-layer population model to a single-layer limit. In a previous paper [Cusulin, C., Iannelli, M., Marinoschi, G. Age-structured diffusion in a multi-layer environment, Nonlinear Anal. Real World Appl. 6(1) (2005) 207–223], we considered a Gurtin–MacCamy type model based on the fact that the population diffuses through a one dimensional habitat, partitioned into n homogeneous layers. In each layer the population has its own age-dependent growth and diffusion parameters, so that within each layer the dynamics is not subject to environmental variations, while changes occur from a layer to another, according to different conditions. Such kind of a model may describe the growth of a population in a fragmented environment, but the same framework may be used to give an approximate view of the population growth and diffusion in a general spatially heterogeneous habitat, because the layer structure may arise by approximation of the original problem. In the present paper we show that this view is actually mathematically sound and justified. In fact, on the basis of the previous results (see [Cusulin, C., Iannelli, M., Marinoschi, G. Age-structured diffusion in a multi-layer environment, Nonlinear Anal. Real World Appl. 6(1) (2005) 207–223]) the approximating problem actually converges and the multi-layer solution may be considered a patch-wise picture of the original problem. 䉷 2006 Elsevier Ltd. All rights reserved. Keywords: Evolution equations; Reaction diffusion; Multi-layer; m-accretive operators; Age structure; Population dynamics
1. Brief introduction and statement of the problem In a previous paper [6] we have considered the problem of an age-structured population diffusing in a one dimensional environment partitioned into n different layers. In each layer the population has its own growth and diffusion parameters, so that within each layer the dynamics is not subject to environmental variations, while changes occur from a layer to another, according to the different conditions. Such kind of a model may describe the growth of a population in a heterogeneous habitat, fragmented into patches where the diffusion coefficient and the vital rates are constant with respect to the spatial variable, and vary only from one patch to another. Our attention here is focused on the one dimensional case, but the approach could be extended to higher dimensions. 夡 This research was supported in part within the FIRB project RBAU01K7M2 “Metodi dell’Analisi Matematica in Biologia, Medicina e Ambiente” of the Italian Ministero Istruzione Università e Ricerca and GAR 22, 2005-2006 of the Romanian Academy. ∗ Corresponding author. Tel.: +39 461 881657; fax: +39 461 881624. E-mail addresses: [email protected] (C. Cusulin), [email protected] (M. Iannelli), [email protected] (G. Marinoschi).
1468-1218/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2006.03.012
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Models for population diffusion in a fragmented habitat have been recently considered by many authors. In particular, [2,3,8,10,11] are concerned with an alternance of patches with different vital rates. In the formulation considered in [6] we have joined diffusion and age structure, proving well-posedness of the problem. Actually, we may note that the framework of the multi-layer model may arise as a stepwise approximation of a model formulated with general diffusion coefficient and vital rates as functions of the space variable. In fact, such functions may be approximated stepwise and a sequence of multi-layer problems is generated as an approximation of the original problem. In the present paper we give a precise formulation of this view and show that it is actually mathematically sound and justified. In fact, on the basis of the previous results (see [6]) we prove that an approximating sequence of problems actually converges and that the multi-layer solution may be considered a patch-wise picture of the original problem. To set a precise formulation of the problem, we denote = (0, a + ) × (y0 , yL ), 0 = {(0, y); y ∈ (y0 , yL )},
a + = {(a + , y); y ∈ (y0 , yL )},
y0 = {(a, y0 ); a ∈ (0, a + )},
yL = {(a, yL ); a ∈ (0, a + )}
and consider the following problem in : jp jp j + + (a, y, S(t, y))p − jt ja jy
a+
p(t, 0, y) =
K(a, y)
(a, y, S(t, y))p(a, t, y) da,
jp jy
=f
in (0, T ) × ,
(1)
(2)
0
S(t, y) =
(a, y, z)p(t, a, z) dz da,
(3)
with the boundary conditions p(0, a, y) = p0 (a, y)
in ,
(4)
K(a, y)
jp =0 jy
on (0, T ) × y0 ,
(5)
K(a, y)
jp =0 jy
on (0, T ) × yL .
(6)
This will be called problem (P ). In the model above a ∈ (0, a + ) is the age, y ∈ (y0 , yL ) is the spatial variable and t ∈ (0, T ) is the time. Moreover, K(a, y), (a, y, x), (a, y, x), are, respectively, the diffusion coefficient, mortality and fertility. Finally, f is some source in the layer and (a, y, z) is a weight function that through the size (3) models how the values of the solution at different points of the environment influence the population at the point y. We note that the problem is actually autonomous, because the way the size S(t, y) depends upon t is actually induced by the solution (see (3)). Besides this, in order to introduce the n-layer structure we denote j = (0, a + ) × (yj −1 , yj ), yj = {(a, yj ); a ∈ (0, a + )},
j = 1, . . . , n, j = 0, . . . , n
C. Cusulin et al. / Nonlinear Analysis: Real World Applications 8 (2007) 887 – 902
889
and consider the following problem for j = 1, 2, . . . , n, on the variables pj (t, a, y) defined on [0, T ] × j ; jpjn jt
jpjn
+
ja
+ nj (a, Sjn (t))pjn − Kjn (a)
j2 pjn jy 2
= fjn
in (0, T ) × j ,
(7)
n pjn (0, a, y) = p0,j (a, y) in j ,
pjn (t, 0, y) = Sjn (t) =
a+ 0
n
a+
k=1
nj (a, Sjn (t))pjn (a, t, y) da,
yk yk−1
0
(8) (9)
nj (a, z)pkn (t, a, z) dz da,
(10)
endowed with the boundary conditions K1n (a)
jp1n =0 jy
on (0, T ) × y0 ,
(11)
Knn (a)
jpnn =0 jy
on (0, T ) × yL ,
(12)
and with the additional conditions on each interface between two layers (for j = 1, 2, . . . , n − 1) pjn = pjn+1 Kjn (a)
jpjn jy
on (0, T ) × yj , = Kjn+1 (a)
jpjn+1 jy
(13) on (0, T ) × yj .
(14)
Here we are using the superscript “n”, since this problem is associated to the partition of the space interval (y0 , yL ) in n layers; we shall refer further to it as problem (P n ). All the functions Kjn , nj , nj , fjn , nj are approximations of the respective functions in problem (P ) and all of them, but nj , are constant within each layer, with respect to the space variable y. Precise assumptions about all these functions and on the way they converge to their respective limits will be stated later within the functional framework that will be set up in the next section. 2. Functional framework The goal of this section is the formulation of a functional framework to treat problems (P ) and (P n ), following the procedure used in [6]. Namely, we want to formulate these problems as abstract Cauchy problems in the space H = L2 (). To this aim we consider the spaces (see [6]) V = H 1 (y0 , yL ),
H = L2 (y0 , yL ),
and, if V is the dual of V , we have V ⊂ H ⊂ V . Then, we consider problem (P ) and define first the operator A0 : D(A0 ) ⊂ L2 (0, a + ; V ) → L2 (0, a + ; V ), on the domain
+
+
a+
D(A0 ) = u ∈ L (0, a ; V ); ua ∈ L (0, a ; V ), u(0, y) = 2
2
0
(a, y, S(y))u(a, y) da ,
(15)
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by the relation
A0 u, = ua , +
((a, y, S(y))u + K(a, y)uy y ) da dy,
(16)
∀ ∈ L2 (0, a + ; V ). By ·, · we denote the duality product between the space L2 (0, a + ; V ) and its dual L2 (0, a + ; V ); we note that a+ f, g = f (a), g(a)V ,V da. 0
We specify that in (16) we set (a, y, z)u(a, z) dz da. S(y) =
(17)
Actually, this is an operator S : H → H defined by (a, y, z)u(a, z) dz da, (Su)(y) =
but for the writing simplicity we skip this notation and use (17). Also we notice that u at a = 0 makes sense, since u ∈ L2 (0, a + ; V ) and ua ∈ L2 (0, a + ; V ) implies u ∈ C([0, a + ]; H ). Finally, we define the operator A : D(A) ⊂ H → H by setting D(A) = {u ∈ D(A0 ), A0 u ∈ H },
Au = A0 u,
∀u ∈ D(A),
and we are led to the Cauchy problem corresponding to (P ) dp + Ap = f dt p(0) = p0 .
a.e. t ∈ (0, T ),
(18) (19)
To place problems (P n ) within the same framework we need first to build new functions on (0, T ) × by a stepwise definition n p1 (t, a, y), y ∈ (y0 , y1 ), n (20) p (t, a, y) = · · · pnn (t, a, y), y ∈ (yn−1 , yL ), n S1 (t), y ∈ (y0 , y1 ), n (21) S (t, y) = · · · Snn (t), y ∈ (yn−1 , yL ), n 1 (a, x), y ∈ (y0 , y1 ), (22) n (a, y, x) = · · · nn (a, x), y ∈ (yn−1 , yL ), n 1 (a, x), y ∈ (y0 , y1 ), n (23) (a, y, x) = · · · nn (a, x), y ∈ (yn−1 , yL ), n 1 (a, z), y ∈ (y0 , y1 ), z ∈ (y0 , yL ), n (24) (a, y, z) = · · · nn (a, z), y ∈ (yn−1 , yL ), z ∈ (y0 , yL ),
C. Cusulin et al. / Nonlinear Analysis: Real World Applications 8 (2007) 887 – 902
n
K (a, y) = p0n (a, y) =
K1n (a), ··· Knn (a),
y ∈ (y0 , y1 ), (25) y ∈ (yn−1 , yL ),
n (a, y), p0,1 ··· n (a, y), p0,n
n
f (a, y) =
891
f1n (a, y), ··· fnn (a, y),
y ∈ (y0 , y1 ), (26) y ∈ (yn−1 , yL ), y ∈ (y0 , y1 ), (27) y ∈ (yn−1 , yL ).
We note that after these definitions S n (t, y) can be rewritten (see (10)) as S n (t, y) = n (a, y, z)p(t, a, z) dz da.
(28)
At this point we stress again that S n (y) is in fact the operator S n : H → H defined by n (S u)(y) = n (a, y, z)u(a, z) da dz.
(29)
Now we may proceed as in the previous definition of the operator A in order to define the following sequence of operators An : D(An ) ⊂ H → H by setting An u = An0 u, on the domain D(An ) = {u ∈ D(An0 ),
An0 u ∈ H },
where An0 : D(An0 ) ⊂ L2 (0, a + ; V ) → L2 (0, a + ; V ) is defined as An0 u, = ua , +
(n (a, y, S n (y))u + K n (a, y)uy y ) da dy,
for all ∈ L2 (0, a + ; V ), on the domain D(An0 ) = u ∈ L2 (0, a + ; V ); ua ∈ L2 (0, a + ; V ), u(0, y) =
a+
n (a, y, S n (y))u(a, y) da .
0
By all these we are led to the Cauchy problem corresponding to (P n ) dpn + An p n = f n dt pn (0) = p0n .
a.e. t ∈ (0, T ),
(30) (31)
Therefore, we can notice that problems (P ) and (P n ) are the same as mathematical form, but the equations have different coefficients. For each problem (P n ) we have existence and uniqueness results obtained in [6] under certain hypotheses and consequently we have similar results for problem (P ) with the corresponding replacements of the functions involved in it. The next section is devoted to a review of the results obtained in [6] and to the introduction of the assumption on our new problem.
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3. Problem hypotheses and review of the existence results Here we summarize the assumptions and briefly explain the way in which the solution to (30)–(31) or (18)–(19) was constructed in [6]: this will be a basis to prove further results about convergence. Actually our assumptions on problems (P n ) repeat those in [6] but also introduce uniformity with respect to the index n. Namely we assume that, for each n, all the functions n (a, y, x), n (a, y, x), n (a, y, z) and K n (a, y) are measurable and we suppose that, for each R > 0, and any x, x ∈ R with |x| R, |x|R, there exist L (R) > 0 and L (R) > 0, such that |n (a, y, x) − n (a, y, x)| L (R)|x − x|,
(32)
|n (a, y, x) − n (a, y, x)| L (R)|x − x|,
(33)
uniformly with respect to a and y. Moreover, we assume that we have 0 n (a, y, x)+
(34)
0 n (a, y, x) with n (a, y, 0) = 0,
(35)
0 n (a, y, z)∞ ,
(36)
0 < K0 K n (a, y) K∞ .
(37)
and
Note that our assumptions directly concern the functions defined on and built in (20)–(27). Of course, such properties may be drawn from those of the components in each layer as done in [6]. Besides this set of assumptions, which includes the basic properties of the functions involved in problem (P n ), we shall impose also the following set of convergence hypotheses to the functions involved in problem (P ). n→∞
n (a, y, x) → (a, y, x) uniformly with respect to a, y and x, n→∞
n (a, y, x) → (a, y, x) uniformly with respect to a, y and x, n→∞
n (a, y, z) → (a, y, z) uniformly with respect to a, y and z, n→∞
K n (a, y) → K(a, y) uniformly with respect to a and y.
(38) (39) (40) (41)
We note that such assumptions (38)–(41) also imply that the same properties (32)–(37) are satisfied by the functions (a, y, x), (a, y, x), (a, y, z), K(a, y) involved in problem (P ) (with the same constants L (R), L (R), + , ∞ , K∞ and K0 ). Thus we have that all the results in [6] can be applied both to (P n ) and (P ). In fact, such results provide the following statement: Theorem 1. Under assumptions (32)–(37), if f n ∈ W 1,1 (0, T ; H ), p0n ∈ D(An ), problem (P n ) has a unique solution pn ∈ W 1,∞ (0, T ; H ) ∩ L∞ (0, T ; D(An )) and it satisfies the estimate T 1 p n (t) 2H p0n 2H + f n (t) 2H dt exp[(2+ a + + 2)t], (42) Kmin 0 with Kmin = min(1, K0 ). The same result holds also for problem (P ). To give an idea of the proof, we will give a few elements which will be useful for further developments. For writing simplicity, in the next explanations we shall refer to the notations for problem (P ).
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The basic step of the proof was to reduce the problem to be solved under the assumptions (32), (33) to another one in which these two assumptions are replaced by global Lipschitz conditions on H for the functions E(u) : H → H and F (u) : H → H defined as follows: E(u) ≡ (a, y, S(y))u(a, y), F (u) ≡ (a, y, S(y))u(a, y), (43) where S(y) = (a, y, z)u(a, y) da dz. In fact, these functions are only locally Lipschitz-continuous and, for our purposes, they were truncated by introducing the approximated functions E(u) for u H N, Nu EN (u) = (44) E for u H > N u H and
FN (u) =
F (u) F
Nu u H
for u H N, for u H > N,
(45)
which are Lipschitz continuous on H , for each N fixed. With these definitions, we consider for each N the corresponding operator AN that coincides with A on the ball of radius N. AN can be proved to be an m-accretive operator so that we have an existence result for the Cauchy dpN + A N pN = f dt
a.e. t ∈ (0, T ),
(46)
pN (0) = p0 ,
(47)
and, since we also can prove the following estimate (see (42)): T 1 2 2 2 pN (t) H N+ = p0 H + f (t) H dt exp[(2+ a + + 2)T ], Kmin 0
(48)
we find that, taking N 2 > N+ , the solution to (46)–(47) is actually a solution of (18)–(19). Moreover, another estimate can be deduced for this problem, that is, T 2 2 2 (f − f )(t) H dt e0 (N)t , (49) p(t) − p(t) H p0 − p0 H + 0
with 0 (N ) = B 2 (N )a + + 2M(N) + 1, where p and p are two solutions corresponding to p0 , f on the one hand and p0 , f on the other hand. Here M(N ) and B(N) are the global Lipschitz constants for EN and FN on H , i.e.,
where
and
EN (u) − EN (u) H M(N ) u − u H ,
(50)
FN (u) − FN (u) H B(N) u − u H ,
(51)
√ √ M(N ) = 2∞ a + yL − y0 L (N )N √ √ B(N ) = ∞ a + yL − y0 L (N )N + + .
In view of the previous sketch of the proof, which holds for the problems (P n ) as well, we conclude that it is sufficient to develop the theory for the truncated problems (because, as we have seen, the solutions to the original problems are in fact the solutions to the truncated ones). n (t) to p (t) and with the operators We mention now that furthermore we shall deal with the truncated solutions pN N n AN and AN associated to them. Since the notation has become too cumbersome, we shall omit the subscript N ; however, the original functions coincide with the truncated ones in the ball of radius N , N being fixed by (48).
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In order to prove the convergence of the solution to (P n ) to the solution to (P ) we will use the following result, given in [4], concerning a sequence of quasi m-accretive operators in a Banach space X, An : D(An ) ⊂ X → X and their “limit” A : D(A) ⊂ X → X, stated in terms of their resolvents. Theorem 2. Let An and A be quasi m-accretive operators and let Un (t) and U(t) be the semigroups generated by −An and −A, respectively. If lim ( + An )−1 g = ( + A)−1 g
n→∞
for every g ∈ D and > 0 , where D =
n 1 D(A
n
) ∩ D(A) and 0 is independent of n, then
lim Un (t)g = U(t)g
n→∞
for every g ∈ D and the limit is uniform on bounded intervals for t. Thus, as seen in the previous theorem, the central point of the study remains the proof of the convergence of the sequence of resolvents. To come to this end, certain intermediate results, including the determination of the closure of the domain of definitions of the operators An and A, some convergence results and estimate settlements for the solutions to the resolvent problems are necessary. The next section is devoted to these preliminaries. 4. Study of the resolvents of the operators An and A Here we consider the m-accretive operators A and An defined in the previous sections under assumptions (32)–(37). We recall that these operators are indeed the truncations AN and AnN defined through the functions FN (·), FNn (·) and n (·), introduced in (44)–(45). We start the study by a density result. EN (·), EN Proposition 1. The domains of the operators A and An satisfy D(A) = D(An ) = H . Proof. Of course, we consider only the operator A, the proof for An being identical. First, let us consider the problem du + A (a)u = 0, da u(0) = u0 ,
(52)
where the linear operator A (a) : V → V is defined by yL A (a)u, V ,V = K(a, y)uy y dy, ∀u ∈ V , ∈ V . y0
(53)
It is known that, if u0 ∈ H , this problem has a solution that we denote by u(a, y; u0 ), such that u ∈ L2 (0, a + ; V ),
ua ∈ L2 (0, a + ; V ),
u(a, ·; u0 ) H u0 H .
(54) (55)
Let now f ∈ H and consider a sequence fn ∈ C ∞ ([0, a + ) × [y0 , yL ]) such that fn → f in H . Let us introduce the mapping Tn : L2 (0, a + ; V ) → H , (Tn g)(a, y) = (na)fn (a, y) + (1 − (na))u(a, y; g), where the mapping : H → H is defined by a+ ( g)(y) = F (g)( , y) d 0
(56)
(57)
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and ∈ C ∞ (R+ ) is a function such that (see [1]) 0 (x)1, (x) = 0
for 0 x 1/2,
(x) = 1 for x 1.
(58)
We note that, according to the definition of F in (43) and to (51), the mapping satisfies √ g − h H B(N) a + g − h H
(59)
and that Tn g ∈ L2 (0, a + ; V ),
(Tn g)a ∈ L2 (0, a + ; V ).
(60)
Moreover, for g, h ∈ H we have yL 1/n Tn g − Tn h 2H = |(1 − (na))u(a, y; g − h)|2 da dy
y0 0 1/n yL 0
y0
|u(a, y; g − h)|2 dy da
B(N)2 a + 1 g − h 2H . g − h 2H n n Here we took into account that for na 1 we have 1 − (na) = 0 and for na 1 we have 1 − (na) 1. Consequently, the integral on [1/n, a + ] vanishes. We used also (55) and (59). In fact, we obtained that Tn is a contraction on H and then the equation Tn g = g has a solution
+ a
gn (a, y) = (na)fn (a, y) + (1 − (na))u a, y;
F (gn ) da .
(61)
0
a+ It can be seen that gn (0, y) = 0 (a, y, S(y))gn (a, y) da and A0 gn ∈ H , so that gn ∈ D(A). Furthermore, from (61) we have 1/n yL 1 |u(a, y; gn )|2 dy da gn 2H fn 2H + 2 y0 0 1 fn 2H + gn 2H n B 2 (N )a + gn 2H , fn 2H + n so that, for n sufficiently large the sequence gn is bounded gn 2H 4 f 2H .
(62)
On the other hand, f (a, y) − gn (a, y) = (na)(f (a, y) − fn (a, y)) + (1 − (na))[f (a, y) − u(a, y; gn )], so that f − gn H f − fn H + and consequently, using (62), f = lim gn n→∞
in H ,
thus proving that D(A) = H .
1/n yL 0
y0
1/2 |f (a, y)| dy da 2
√ B(N ) a + + gn H √ n
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We notice that we have also the following additional result Proposition 2. Let f ∈ L1 (0, T ; H ),
p0 ∈ D(A) = H ,
(63)
then there exists a weak (mild) solution to the Cauchy problem (46)–(47) satisfying t 1 1 p(t) − x 2H p(s) − x 2H − (f − Ax, p() − x) d, 2 2 s ∀x ∈ H and ∀s, 0 s t T . In fact, the operator A, being m-accretive, generates a non-linear semigroup of quasi contractions on D(A) = H . We also recall that in [6] we considered the resolvent problem u + Au = g, where g ∈ D(A) = H , which can be explicited as du + AV (a)u = g a.e. a ∈ (0, a + ), da a+ (a, y, S(y))u(a, y) da. u(0) = u +
(64) (65)
0
Here the operator AV (a) : V → V is defined by yL (K(a, y)uy y + (a, y, S(y))u ) dy, AV (a)u, V ,V = y0
∀ ∈ V , so that problem (64)–(65) may be equivalently written as ua , + ( u + K(a, y)uy y + (a, y, S(y))u − g) da dy = 0,
(66)
∀ ∈ L2 (0, a + ; V ). In [6] it was shown that problem (64)–(65) has a unique solution u ∈ L2 (0, a + ; V ) with ua ∈ L2 (0, a + ; V ), for B 2 (N )a + + (B 2 (N )a + )2 + 4M 2 (N ) . > c = 2 We now need some estimates that we prove in the following: Proposition 3. The solution to the resolvent problem (64)–(65) associated to problem (P ) satisfies the estimates u 2H + u(a + , ·) 2H + K0 u 2L2 (0,a + ;V ) g 2H ,
(67)
ua L2 (0,a + ;V )
(68)
+ M(N) K∞ +√ + 1 g H , √ K0 0
[2 − (1 + B(N)2 a + + M(N ) + K0 )] u − u 2H + u(a + , ·) − u(a + , ·) 2H + K0 u − u 2L2 (0,a + ;V ) g − g 2H , where u and u are two solutions corresponding to g and g and 0 ≡ max(1 + 2+ a + + K0 , c ).
(69)
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897
Proof. Let g ∈ H . In order to prove estimate (67) we multiply Eq. (64) by u and integrate over . We have 1 yL 2 + 1 yL 2 u (a , y) dy − u (0, y) dy + K(a, y)u2y da dy g H u H . u 2H + 2 y0 2 y0 But
yL
y0
u2 (0, y) dy 2+ a + u 2H ,
(70)
whence we get [2 − (1 + 2+ a + + K0 )] u 2H + u(a + , ·) 2H + K0 u 2L2 (0,a + ;V ) g 2H . This implies that for > 0 we obtain (67) as claimed. Concerning (69), we take g, g ∈ H , multiply the equation (u − u) + (Au − Au) = g − g by u − u and we integrate over . Since yL (u − u)2 (0, y) dy B 2 (N )a + u − u 2H , y0
(71)
after similar calculations as before we get (69) as claimed. Finally, for estimating the norm of ua we use the definition ua L2 (0,a + ;V ) =
sup ∈L2 (0,a + ;V ), 1
|ua ()|
(72)
and calculate ua () from (66). We have
+
a
|ua ()| = ua (a), (a)V ,V da
0
=
(− u − Kuy y − (a, y, S(y))u + g) da dy
( u H + K∞ u L2 (0,a + ;V ) + M(N ) u H + g H ) L2 (0,a + ;V ) K∞ + M(N ) +√ + 1 g H L2 (0,a + ;V ) √ K0 0 and we get (68).
Before concluding this section we note that the results of Propositions 2 and 3 hold also for the sequence of operators An and all the estimates are uniform with respect to n. In particular, we have the following: Proposition 4. For each n, the sequence un = ( + An )−1 ∈ D(An ) satisfies the estimates un 2H + un (a + , ·) 2H + K0 un 2L2 (0,a + ;V ) g 2H , 2 − (1 + B 2 (N )a + + M(N ) + K0 ) un − un 2H + un (a + , ·) − un (a + , ·) 2H + K0 un − un 2L2 (0,a + ;V ) g − g 2H , and una L2 (0,a + ;V )
+ M(R) K∞ +√ + 1 g H , √ K0 0
for > 0 , with the same a + , + , ∞ , K0 , K∞ , B(N ), M(N ) and 0 for all n.
(73)
(74)
(75)
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5. Convergence of the semigroup In this section we will use Theorem 2 to prove the convergence of the sequence of the semigroups generated by −An to the semigroup generated by −A. Namely we will prove convergence for the Cauchy problems with null terms f and f n . We give first an intermediate convergence result. Lemma 1. Let (34)–(37) and (38)–(41) and assume that un → u strongly in H and weakly in L2 (0, a + ; V ). Then F n (un ) → F (u) strongly in H ,
(76)
E n (un ) → E(u) strongly in H ,
(77)
K n uny → Kuy
(78)
weakly in H .
Proof. For proving (76) we first note that, for any fixed u ∈ H and > 0, for n sufficiently large we have |F n (u)(a, y) − F (u)(a, y)| |n (a, y, S n (y)) − (a, y, S n (y))||u(a, y)| + |(a, y, S n (y)) − (a, y, S(y))||u(a, y)| |u(a, y)| + L (N )|S n (y) − S(y)||u(a, y)| |u(a, y)|(1 + L (N )), where we have used (38) and (40). Thus we have lim F n (u) = F (u),
n→∞
∀u ∈ H ,
and (76) is proved because F n (un ) − F (u) H F n (un ) − F n (u) H + F n (u) − F (u) H B(N) un − u H + F n (u) − F (u) H . Of course, (77) can be proved in the same way. Concerning (78) we note that, for any ∈ H and > 0, for n sufficiently large we have
(K n un − Kuy ) da dy y
n n n
|K − K||uy ||| da dy + K(uy − uy ) da dy
n n
ε uy H H + K(uy − uy ) da dy
.
Thus, since
uny H
is bounded and the function K(a, y)(a, y) belongs to H , we get (78).
Now, in view of Theorem 2 we prove covergence of the resolvents: Theorem 3. Let g ∈ H and assume all properties (34)–(37) and the convergence hypotheses (38)–(41). Then lim ( + An )−1 g = ( + A)−1 g
n→∞
in H .
(79)
Proof. Let us fix g ∈ H and denote un = ( + An )−1 g. We have to prove that un has a limit and this is exactly ( + A)−1 g. Assumptions (34)–(37) imply immediately (73)–(75), as specified in Proposition 4. Therefore we deduce that the sequence un is bounded in H ∩ L2 (0, a + ; V ), una is bounded in L2 (0, T ; V ) and un (a + ) lies in a bounded subset of H .
C. Cusulin et al. / Nonlinear Analysis: Real World Applications 8 (2007) 887 – 902
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Thus, extracting a subsequence we have the following convergence results: un → u weakly in L2 (0, a + ; V ),
(80)
una → ua
(81)
weakly in L2 (0, a + ; V ),
which should be understood in the sense of distributions and un (a + ) →
weakly in V .
(82)
From the first two relationships and since V is compact in H , we get (see [9]) that un → u
strongly in L2 (0, a + ; H ) = H .
(83)
Now, we may pass to the limit in the equality una , + ( un + K n (a, y)uny y + n (a, y, S n (y))un − g) da dy = 0,
∀ ∈ L2 (0, a + ; V ) and obtain, using (80), (81) and Lemma 1, that ua , + ( u + K(a, y)uy y + (a, y, S(y))u − g) da dy = 0,
∀ ∈ L2 (0, a + ; V ). Now it is obvious that u is the solution to (66), which is the resolvent problem associated to (P ). We can also prove that un (0) → u(0)
strongly in H
and un (a + ) → u(a + ) weakly in V . Indeed, we have
un (0) − u(0) 2H =
yL y0
+
2
a
(F n (un ) − F (u)) da dy
0
+
a F n (un ) − F (u) 2H a + B 2 (N ) un − u 2H . Then we can write un (a + , y) = un (0) +
a+ 0
una (a, y) da
and since we have just proved that un (0) → u(0) strongly in H it remains to show only that a+ 0 ua (a, y) da weakly in V . The latter follows due to (81), since we have for any ∈ V that
yL
y0
a+
(y) 0
una (a, y) da dy
yL
y0 yL
=
→ since ∈ L2 (0, a + ; V ).
y0
a+ 0 a+ 0
a+ 0
una (a, y) da →
(y)una (a, y) da dy (y)ua (a, y) da dy =
yL y0
a+
(y) 0
ua (a, y) da dy,
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6. Convergence for the Cauchy problems We now move on to the proof of the final results concerning problems (P n ) and (P ). First we take f ∈ L1 (0, T ; H ) and p0 ∈ H being the same as in problem (P ) and consider the problem (P n ) in which the free term and initial data are exactly f and p0 . d pn n = f , + An p dt n (0) = p0 . p
(84)
n ). Then we have: We shall call it (P Theorem 4. Let N be fixed, f ∈ L1 (0, T ; H ), p0 ∈ H and assume (34)–(37) and (38)–(41). Then the solution to n ) tends to the solution to problem (P ), i.e. problem (P n (t) → p(t) p
uniformly for t ∈ [0, T ].
(85)
Proof. Assume first that f ∈ W 1,1 (0, ∞; H ) and according to [7], define the following operator in the space X = H × L1 (0, ∞; H ): An : D(An ) = D(A) × W 1,1 (0, ∞; H ) ⊂ X → X by An (u, ) =
An u − (0) , −
(86)
for all (u, ) ∈ D(A) × W 1,1 (0, ∞; H ). If we denote n (t) p n , P (t) = (t)
(87)
where ∈ W 1,∞ (0, T ; W 1,1 (0, ∞; H )) is defined by (t)(s) = f (t + s),
∀s ∈ (0, ∞),
(88)
it follows that the problem dPn (t) + An Pn (t) = 0 dt p0 Pn (0) = f (s)
a.e. t ∈ (0, T ), (89)
n ). is equivalent to (P Indeed, the second component of (89) gives the equations j(t, s) j(t, s) − = 0, jt js
(0, s) = f (s),
which are necessarily satisfied by the solution (t, s) = f (t + s). Writing the first component in (89) we get d pn n (t) − (t)(0) = 0 (t) + An p dt
a.e. t ∈ (0, T ),
n ). and, since (t)(0) = f (t), we obtain the equation and the initial condition for problem (P
C. Cusulin et al. / Nonlinear Analysis: Real World Applications 8 (2007) 887 – 902
901
Of course, the same construction performed above can be repeated for problem (P ) and in the same space X we can define an operator Au − (0) , A(u, ) = − on the domain D(A) = D(A) × W 1,1 (0, ∞; H ). Further, according to the results given in [5,7] it turns out that the operators An and A are quasi m-accretive on X. Also, by Proposition 1 we can deduce immediately that D(An ) = D(A) = X. Finally, we have that the resolvent of the operator An tends to the resolvent of the operator A. To show this we fix ( g ) ∈ X and, since the operator An is quasi m-accretive, for sufficiently large we may consider n
( un ) ∈ D(An ) × W 1,1 (0, ∞; H ), given by n u n −1 g . = ( I + A ) n
This means that we have un + An un − n (0) = g, n − (n ) = . The solution to the second equation is s n s − (s) = e C− e () d
for > 0
0
and the constant C should be determined such that n ∈ W 1,1 (0, ∞; H ). Hence we impose that lims→∞ n (s) = 0 and it follows that ∞ e− () d. n (s) = e s s
We notice that this solution does not depend on n, i.e., n (s) = (s) and introducing it in the first equation we obtain ∞ n n n e− () d. u + A u = g + Since
∞ 0
0
e− () d
un n
=
∈ H we deduce by Theorem 3 that
∞ ∞ −1 g + 0 e− () d ( I + An )−1 g + 0 e− () d n→∞ ( I + A) u → = . ∞ − ∞ − s s e s e () d e s e () d
It is obvious that u g = ( + A)−1 . To conclude, resuming Theorem 2 we get then that the semigroup generated by the operator −An corresponding to n ) tends to the semigroup generated by −A, i.e. problem (P n (t) n→∞ p(t) p Pn (t) = → = P(t), (t) (t) uniformly for t ∈ [0, T ].
Now we take f ∈ L1 (0, T ; H ), extend it by 0 to (T , ∞) and consider a sequence fm ∈ W 1,1 (0, ∞; H ), such m→∞ that fm → f in L2 (0, ∞; H ). It is obvious that the result of Theorem 4 is preserved, on the basis of estimate (49) (written for problem (P n )) and it is also true for any f ∈ L2 (0, T ; H ).
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The last theorem gathers all the results obtained up to now and reaches the goal of our paper. Theorem 5. Let N be fixed, f n ∈ L2 (0, T ; H ), p0 ∈ H and assume (34)–(37), (38)–(41) and the convergence relationships fn → f
strongly in L2 (0, T ; H ),
(90)
p0n → p0
strongly in L2 (0, T ; H ).
(91)
Then lim p n (t) = p(t),
n→∞
uniformly for any t ∈ [0, T ],
(92)
where p n (t) and p(t) are the solutions to problems (P n ) and (P ), respectively. Proof. Consider the problem (P n ) corresponding to two sets of initial data and free terms, respectively, to f, p0 and n ). f n , p0n . The first one is in fact problem (P We extend f n to (0, ∞) by setting f n (t) = 0 for t ∈ [T , ∞). By (49), (90) and (91) we get that n (t) 2H → 0 pn (t) − p
as n → ∞. n→∞
n (t) → p(t), so finally we obtain that the solution p n (t) to problem (P n ) tends to the By Theorem 4 we have that p solution to problem (P ) uniformly for t ∈ [0, T ]. References [1] V. Barbu, M. Iannelli, The semigroup approach to non-linear age-structured equations, Rend. Istit. Mat. Univ. Trieste (Suppl. vol. XXVIII) (1997) 59–71. [2] H. Berestycki, F. Hamel, L. Roques, Analysis of the periodically fragmented environment model: I—species persistence, J. Math. Biol. 51 (2005) 75–113. [3] H. Berestycki, F. Hamel, L. Roques, Analysis of the periodically fragmented environment model: II—Biological invasions and pulsating travelling fronts, J. Math. Pures Appl. 84 (2005) 1101–1146. [4] H. Brezis, A. Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces, J. Funct. Anal. 9 (1972) 63–74. [5] M.G. Crandall, A. Pazy, An approximation of integrable functions by step functions with an application, Proc. Am. Math. Soc. 76 (1) (1979) 74–80. [6] C. Cusulin, M. Iannelli, G. Marinoschi, Age-structured diffusion in a multi-layer environment, Nonlinear Anal. Real World Appl. 6 (1) (2005) 207–223. [7] C.M. Dafermos, M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups, J. Functional Anal. 13 (1973) 97–106. [8] N. Kinezaki, K. Kawasaki, F. Takasu, N. Shigesada, Modeling biological invasion into periodically fragmented environments, Theor. Popul. Biol. 64 (2003) 291–302. [9] J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969. [10] N. Shigesada, K. Kawasaki, Biological invasions: theory and practice. Oxford Series in Ecology and Evolution, Oxford University Press, Oxford, 1997. [11] N. Shigesada, K. Kawasaki, E. Teramoto, Traveling periodic waves in heterogeneous environments, Theor. Popul. Biol. 30 (1986) 143–160.