Age-dependent diffusive Lotka-Volterra type systems

Transcription

1 Age-dependent diffusive Lotka-Volterra type systems M. Delgado and A. Suárez 1 Dpto. Ecuaciones Diferenciales y Análisis Numérico Fac. Matemáticas, C/ Tarfia s/n C.P. 4112, Univ. Sevilla, Spain addresses: madelgado@us.es and suarez@us.es Abstract In this paper it is shown that the sub-supersolution method works for age-dependent diffusive nonlinear systems with non-local initial conditions. As application, we prove the existence and uniqueness of positive solution for a kind of Lotka-Volterra systems, as well as the blow-up in finite time in a particular case. Key Words. Sub and supersolutions, Lotka-Volterra systems, backward problem, blowup. AMS Classification. 35B5, 35K57, 35L6, 92D25 1 Introduction The introduction of the age-structure in the population dynamic models supposes a considerable advance insofar as it permits the dependence of the age of parameters so sensitive to it as the birth and mortality rates are. From the mathematical point of view, the combination of the equation and, mainly, the nonlocal initial condition for the age presents many interesting and nontrivial questions. After, this structure has been exploited to describe the evolution of a population divided in two sub-populations in, for instance, the frame of the epidemics theory; in this case, the age structure is rather a contagion-time structure. 1 Supported by the Spanish Ministry of Science and Technology under Grant BFM

2 2 M. Delgado and A. Suárez A former step is the introduction the age structure in the interaction of species with or without diffusion. The first attempts were the interaction of two species in which one of them has an age-structure and the other one has a structure independent of the age, see [8], [13], or only numeric approaches when the age-structure is considered for the two species, [15]. But as long as we know the theoretical age-dependent problem for the two species has not been tackled. Because the interactions between the species are nonlinear, it is necessary to solve some problems for one age-dependent equation with a nonlinear reaction term. For this kind of problems, unsolved also in our knowledge, we proved the validity of the sub-supersolutions method (see [9]). So, it is the moment to profit it to approach the problem of the interaction of two species with age-structure, extending the sub-supersolutions method to systems and checking the results when the interaction of the species are the classical competition, preypredator and symbiotic. Our main goal in this paper is the study of the application of the sub-supersolutions method to systems age depending. The model we present perhaps it is not the most realistic possible, but it allows us to check the difficulties and advantages of the application on the cited method. We analyze the existence and uniqueness of positive solution of the following agedependent diffusive Lotka-Volterra systems u t + u a u + µ 1 (x, a, t)u = u(λ u + bv) in Ω O, v t + v a v + µ 2 (x, a, t)v = v(ν v + cu) in Ω O, u(x, a, t) = v(x, a, t) = on Ω O, u(x, a, ) = u (x, a), v(x, a, ) = v (x, a) in Ω (, A ), u(x,, t) = v(x,, t) = β 1 (x, a, t)u(x, a, t)da in Ω (, T ), β 2 (x, a, t)v(x, a, t)da in Ω (, T ), (1.1) where Ω IR N is a bounded domain with smooth boundary Ω, A, T >, O := (, A ) (, T ), λ, ν, b, c IR and

3 Age-dependent Lotka-Volterra diffusive systems 3 (H1) µ i C (Ω [, A ) [, T ]), µ i (x, a, t) t < t < A, x Ω, lim a A µ i (x, a t + τ, τ)dτ = +, A < t < T, x Ω, a lim a A µ i (x, τ, t a + τ)dτ = +. (H2) β i L (Ω O), β i (x, a, t). We will denote β i := ess sup (x,a,t) Ω O β i (x, a, t). (H3) u, v L 2 (Ω (, A )). System (1.1) models the behavior of two species with densities u(x, a, t) and v(x, a, t) of age a > at time t > and at position x Ω, which cohabit in Ω. Here µ i and β i denote the natural death and fertility rates of each species, respectively. The species are interacting in three different ways: if b, c < they are competing, if b, c > cooperating and finally, if for instance b > and c <, u represents the predator and v the prey. In this context, b and c represent the interaction rates between the species and, finally, λ and ν are the growth rates of the species, and they are considered as parameters. We note that hypothesis (H1) assures that the solutions u and v vanish at a = A (see [11]), and so A is the highest age attained by the individuals in the populations. In another way, the positivity of the mortality rates of species is a natural but mathematically unimportant condition as can be seen with the change of variables w = e kt u, z = e kt v for k > big enough. In our knowledge, diffusive with age dependence nonlinear systems have not been analyzed deeply previously. In [1], the local existence for a system is studied when the nonlinearities satisfy some conditions out of our setting. In [16] and [17] two prey-predator systems, including the total populations in the model, are analyzed by means a fixed point theorem. In this paper, we prove mainly the following result: a) In the competition (b, c < ), prey-predator (bc < ) and weak cooperating (b, c > and bc < 1) cases, there exists a unique positive solution for all λ, ν IR for all time T >. b) In the strong cooperating case (b, c > and bc > 1) there exists a value λ such that for λ, ν > λ the solution blows up in finite time.

4 4 M. Delgado and A. Suárez Observe that the results are in concordance with the ones obtained in the case of not age dependence, see for instance [14]. In order to prove these results, first we show that the sub-supersolution method works for this kind of systems, generalizing to systems the result obtained in the scalar case in [9]. Then, we find appropriate sub-supersolutions in each case. However, the proof of the blow-up result is more involved. In fact, to solve the eigenvalue problem associated to (1.1), we define a compact operator whose principal eigenfunction is transformed to obtain the principal eigenfunction of our problem. We study the adjoint of the former operator, see [6], [2], [4], [3], that leads us to a backward problem whose solution has a similar transformation to build a function, ϕ, solution of a new backward problem also, which is used to prove the blow-up expected. In Section 2 we build a sub-supersolution method. Section 3 is devoted to study the existence and uniqueness of positive solutions of (1.1). Finally, in Section 4 we show that in the strong cooperating case the solution blows up in finite time. 2 The sub-supersolution method We consider the system u t + u a u + µ 1 (x, a, t)u = f(x, a, t, u, v) in Ω O, v t + v a v + µ 2 (x, a, t)v = g(x, a, t, u, v) in Ω O, u(x, a, t) = v(x, a, t) = on Ω O, u(x, a, ) = u (x, a), v(x, a, ) = v (x, a) in Ω (, A ), u(x,, t) = v(x,, t) = β 1 (x, a, t)u(x, a, t)da, in Ω (, T ), β 2 (x, a, t)v(x, a, t)da, in Ω (, T ). (2.1) Definition 2.1 A couple (u, v) is a solution of (2.1) if u, v : Ω O IR are measurable functions such that u, v L 2 (O; H 1 (Ω)) and verify u t + u a + µ 1 u L 2 (O; H 1 (Ω)), v t + v a + µ 2 v L 2 (O; H 1 (Ω)) f(,,, u, v) L 2 (O; H 1 (Ω)), g(,,, u, v) L 2 (O; H 1 (Ω))

5 Age-dependent Lotka-Volterra diffusive systems 5 and for every w L 2 (O; H 1 (Ω)) the following equalities hold u t + u a + µ 1 u, w dadt + u wdxdadt = f(, a, t, u, v), w dadt O Ω O O v t + v a + µ 2 v, w dadt + v wdxdadt = g(, a, t, u, v), w dadt O Ω O where, stands for the duality between H 1 (Ω) and H 1 (Ω). Moreover, the initial conditions for the time and for the age must be verified in L 2 (Ω (, A )) and L 2 (Ω (, T )) respectively. O Definition 2.2 Let u, v L 2 (O; L 2 (O Ω)), with u v. We define the interval [u, v] as [u, v] := {z L 2 (O; L 2 (O Ω)) : u z v} We define now the suitable concept of sub-supersolutions. Definition 2.3 Two couples (u, u), (v, v) (L 2 (O, H 1 (Ω)) 2 are a pair of sub-supersolutions of (2.1) if u u, v v in Ω O f(,,, u, v), f(,,, u, v) L 2 (O, (H 1 (Ω)) ), v [v, v] g(,,, u, v), g(,,, u, v) L 2 (O, (H 1 (Ω)) ), u [u, u] Moreover, u must verified (and analogous conditions for the other functions) a) u t + u a + µ 1 u L 2 (O; (H 1 (Ω)) ). b) O u t + u a + µ 1 u, w dadt + u wdxdadt f(, a, t, u, v), w dadt Ω O O for each w L 2 (O, H 1 (Ω)), w and for each v [v, v]. c) u(x, a, t) on Ω O. d) u(x,, t) β 1 (x, a, t)u(x, a, t)da, for (x, t) Ω (, T ). e) u(x, a, ) u (x, a), for (x, a) Ω (, A ). We will try to establish a sub-supersolution theorem, which will give us furthermore a priori bounds of the solutions in many cases. The result is

6 6 M. Delgado and A. Suárez Theorem 2.4 Suppose (H1), (H2), (H3) and that there exists L > such that f(x, a, t, r 1, s 1 ) f(x, a, t, r 2, s 2 ) L( r 1 r 2 + s 1 s 2 ) g(x, a, t, r 1, s 1 ) g(x, a, t, r 2, s 2 ) L( r 1 r 2 + s 1 s 2 ) a.e. (x, a, t) Ω O (2.2) r 1, r 2 [u, u ], s 1, s 2 [v, v ], being u := ess inf Ω O u(x, a, t), u := ess sup Ω O u(x, a, t), v := ess inf Ω O v(x, a, t), v := ess sup Ω O v(x, a, t). Then, (2.1) possesses a unique solution (u, v) such that (u, v) [u, u] [v, v]. Proof. We define f(x, a, t, u, v) = f(x, a, t, r, s), g(x, a, t, u, v) = g(x, a, t, r, s) being u if u u v if v v r = u if u u u s = v if v v v u if u u v if v v We are going to prove that problem (2.1) with f and g instead of f and g has a unique solution and that this solution belongs to the rectangle [u, u] [v, v]. If we perform the change of variables u = e αt w, v = e αt z, the problem to solve is w t + w a w + (µ 1 (x, a, t) + α)w = e αt f(x, a, t, e αt w, e αt z) in Ω O, z t + z a z + (µ 2 (x, a, t) + α)z = e αt g(x, a, t, e αt w, e αt z) in Ω O, w(x, a, t) = z(x, a, t) = on Ω O, w(x, a, ) = u (x, a), z(x, a, ) = v (x, a) in Ω (, A ), w(x,, t) = z(x,, t) = β 1 (x, a, t)w(x, a, t)da, in Ω (, T ), β 2 (x, a, t)z(x, a, t)da in Ω (, T ). (2.3)

7 Age-dependent Lotka-Volterra diffusive systems 7 We consider the space E := L 2 (O, H 1 (Ω)) L 2 (O, H 1 (Ω)) with the norm (u, v) E = ( u 2 L 2 (O,H 1 (Ω)) + v 2 L 2 (O,H 1 (Ω)) )1/2 ; it is well known that w 2 L 2 (O,H 1 (Ω)) := [11]) the norm in H 1 (Ω) O w(x, a, t) 2 H 1 (Ω) dadt and we will use (following w 2 α := w 2 L 2 (Ω) + 1 α w 2 L 2 (Ω) N for some α >. We define the map Λ : E E; (w, z) ( w, z) being w t + w a w + (L + µ 1 (x, a, t) + α) w = e αt f(x, a, t, e αt w, e αt z) + Lw in Ω O, z t + z a z + (L + µ 2 (x, a, t) + α) z = e αt g(x, a, t, e αt w, e αt z) + Lz in Ω O, w(x, a, t) = z(x, a, t) = on Ω O, w(x, a, ) = u (x, a), z(x, a, ) = v (x, a) in Ω (, A ), w(x,, t) = z(x,, t) = β 1 (x, a, t)w(x, a, t)da in Ω (, T ), β 2 (x, a, t)z(x, a, t)da in Ω (, T ). (2.4) These are two uncoupled linear problems and it is easy to see that the second members of the equations are in L 2 (O Ω). Hence, the operator is well defined. We denote := [e αt u, e αt u] [e αt v, e αt v], a.e. (x, a, t) O Ω and check that the restriction Λ (which we will denote the same) verifies that Λ :. In fact, let (w, z) and ( w, z) := Λ(w, z); if we pose w = e αt u w, it is easy to

8 8 M. Delgado and A. Suárez see that w t + w a w + (L + α + µ 1 )w e αt (f(x, a, t, u, v) f(x, a, t, e αt w, e αt z)) + L(e αt u w), w (x, a, t), on Ω O, w (x, a, ) in Ω (, A ), w (x,, t) on Ω (, T ), for any v [v, v]. We choose v = e αt z and by the property (2.2), w t + w a w + (L + α + µ 1 )w. From the maximum principle for these linear problems (see Lemma 2.4 in [9]) we deduce that w, i.e, w e αt u. In a analogous way the other inequalities can be proved. We claim that with a suitable choice of α, the map is contractive. Let (w 1, z 1 ), (w 2, z 2 ) E and (w, z ) := Λ(w 1, z 1 ) Λ(w 2, z 2 ) = ( w 1 w 2, z 1 z 2 ) We have to prove that K, < K < 1 : (w, z ) E K (w 1 w 2, z 1 z 2 ) E But we know w t + w a w + (L + µ 1 (x, a, t) + α)w = e αt ( f(x, a, t, e αt w 1, e αt z 1 ) f(x, a, t, e αt w 2, e αt z 2 )) + L(w 1 w 2 ), z t + z a z + (L + µ 2 (x, a, t) + α)z = e αt ( g(x, a, t, e αt w 1, e αt z 1 ) g(x, a, t, e αt w 2, e αt z 2 )) + L(z 1 z 2 ), w (x, a, t) = z (x, a, t) = on Ω O, w (x, a, ) =, z (x, a, ) = in Ω (, A ), w (x,, t) = z (x,, t) = β 1 (x, a, t)(w 1 w 2 )(x, a, t)da in Ω (, T ), β 2 (x, a, t)(z 1 z 2 )(x, a, t)da in Ω (, T ). (2.5)

9 Age-dependent Lotka-Volterra diffusive systems 9 Then, we take < A < A and the test function φ := w χ (,A ); we multiply the first equation by φ, integrate on Ω O ; O := (, A ) (, T ), apply the integration by parts formula and it results 1 2 β2 1A (w 1 w 2 ) 2 (x, a, t)dxdadt + w 2 dxdadt+ Ω O Ω O (L + µ 1 (x, a, t) + α) w 2 dxdadt L w 1 w 2 w dxdadt+ Ω O O e αt f(x, a, t, e αt w 1, e αt z 1 ) f(x, a, t, e αt w 2, e αt z 2 ), w. O But, applying (2.2) and the trivial inequality (2m + n)p 3m 2 + 2n p2, m, n, p IR it yields L (w 1 w 2 )w dxdadt+ O e αt f(x, a, t, e αt w 1, e αt z 1 ) f(x, a, t, e αt w 2, e αt z 2 ), w O e αt f(x, a, t, e αt w 1, e αt z 1 ) f(x, a, t, e αt w 2, e αt z 2 ) L 2 (Ω) w L 2 (Ω)dadt+ O e αt L w 1 w 2 L 2 (Ω) w L 2 (Ω)dadt O (2L w 1 w 2 L 2 (Ω) + L z 1 z 2 L 2 (Ω)) w L 2 (Ω)dadt O 3L 2 w 1 w 2 2 dxdadt + 2L 2 z 1 z 2 2 dxdadt + α w 2 Ω O Ω O 2 αdadt O if we choose α > 1. So, 1 2 β2 1A Ω O w 1 w 2 2 dxdadt + w 2 dxdadt+ Ω O (L + µ 1 (x, a, t) + α) w 2 dxdadt Ω O 3L 2 w 1 w 2 2 dxdadt + 2L 2 z 1 z 2 2 dxdadt + α w 2 Ω O Ω O 2 αdadt O and, since µ 1, w 2 L O 2 (Ω) dadt + (L + α) w 2 L O 2 (Ω) dadt α w 2 2 αdadt O (3L β2 1A ) O w 1 w 2 2 L 2 (Ω) dadt + 2L2 O z 1 z 2 2 L 2 (Ω) dadt.

10 1 M. Delgado and A. Suárez It is easy to see that and taking A A, Analogously, we can obtain Therefore α w 2 2 αdadt (3L O 2 β2 1A ) (w 1 w 2, z 1 z 2 ) 2 E α 2 w 2 L 2 (O,H 1 (Ω)) (3L β2 1A ) (w 1 w 2, z 1 z 2 ) 2 E α 2 z 2 L 2 (O,H 1 (Ω)) (3L β2 2A ) (w 1 w 2, z 1 z 2 ) 2 E α 2 (w, z ) 2 E (3L max(β2 1, β 2 2)A ) (w 1 w 2, z 1 z 2 ) 2 E and we can choose α > 1 such that the map Λ be contractive. Remark 2.5 The result can be extended for any number of equations. 3 Applications: The Lotka-Volterra models In this section we apply the sub-supersolution method to the systems u t + u a u + µ 1 (x, a, t)u = u(λ u + bv) in Ω O, v t + v a v + µ 2 (x, a, t)v = v(ν v + cu) in Ω O, u(x, a, t) = v(x, a, t) = on Ω O, u(x, a, ) = u (x, a), v(x, a, ) = v (x, a) in Ω (, A ), u(x,, t) = v(x,, t) = β 1 (x, a, t)u(x, a, t)da in Ω (, T ), β 2 (x, a, t)v(x, a, t)da in Ω (, T ), (3.1) where µ i, β i satisfy (H1), (H2), λ, ν, b, c IR and, instead of (H3), we assume (H4) u, v L (Ω (, A )) and u, v.

11 Age-dependent Lotka-Volterra diffusive systems 11 Before studying (3.1), we need to analyze the logistic equation u t + u a u + µ(x, a, t)u = u(λ(x, a, t) u) in Ω O, u(x, a, t) = on Ω O, u(x, a, ) = u (x, a) in Ω (, A ), u(x,, t) = β(x, a, t)u(x, a, t)da in Ω (, T ), (3.2) where µ, β and u satisfy (H1), (H2) and (H4), respectively and λ L (Ω O). Equation (3.2) was studied in [9] under more restrictive conditions on the data. We present the main result for reader s convenience. Proposition 3.1 There exists a unique positive solution of (3.2), denoted by Θ [λ,µ,β]. Moreover, Θ [λ,µ,β] is bounded in L (Ω O). Proof: We are going to find a sub-supersolution of (3.2). Take B > such that β(x, a, t) B, and consider consider the problem u t + u a u = λu in Ω O, u(x, a, t) = on Ω O, u(x, a, ) = u (x, a) in Ω (, A ), u(x,, t) = B u(x, a, t)da in Ω (, T ), (3.3) where λ = ess sup (x,a,t) Ω O λ(x, a, t). If we denote by ω λ the unique positive solution of (3.3) which is bounded, see [9], then, (, ω λ ) is a sub-supersolution of (3.2), so that there exists a unique positive solution in [, ω λ ], because the lipschitzianity of u(λ u) in this bounded interval. But any possible solution of (3.2), u, is a subsolution of (3.3) and so u ω λ. Thus, u = v and the uniqueness follows. Now, we are ready to state and prove the main result: Theorem 3.2 a) Competition case: Assume that b, c <. Then, there exists a unique positive solution of (3.1). b) Prey-predator case: Assume that bc <. Then, there exists a unique positive solution of (3.1).

12 12 M. Delgado and A. Suárez c) Weak cooperating case: Assume that b, c > and bc < 1. Then, there exists a unique positive solution of (3.1). Proof: We have to build a sub-supersolution couple in each case. Assume that b, c <. Then, it is clear that (u, v) = (, ), (u, v) = (Θ [λ,µ1,β 1 ], Θ [ν,µ2,β 2 ]) is a sub-supersolution of (3.1). On the other hand, if (u, v) is solution of (3.1), then u is sub-solution of (3.2) with µ = µ 1, β = β 1 and λ IR. Then, u Θ [λ,µ1,β 1 ], and analogously, v Θ [ν,µ2,β 2 ]. So, any solution belongs to a bounded interval where the second members of (3.1) are Lipschitz. This concludes the uniqueness. Assume now that bc <, for instance b > and c <. Then, again it is not hard to show that the following couple is sub-supersolution of (3.1): (u, v) = (, ), (u, v) = (Θ [λ+bθ[ν,µ2,β 2 ],µ 1,β 1 ], Θ [ν,µ2,β 2 ]) Observe that u is well defined because Θ [ν,µ2,β 2 ] is bounded. The uniqueness follows in the same way than in the competition case. that Finally, assume that b, c > and bc < 1. Take one function m C[, A ], B > such m(a) µ i (x, a, t), β i B, i = 1, 2. For this choice, let r m the root of the equation Denote a 1 = B exp (ra m(s)ds)da. a g(a) := exp ( r m a m(s)ds). and ˆλ 1 the principal eigenvalue of in ˆΩ, a domain such that Ω ˆΩ. Now, consider the pair (u, v) = (, ), (u, v) = (K 1 g(a) ˆϕ 1, K 2 g(a) ˆϕ 1 ), where K 1, K 2 > and ˆϕ 1 is a positive eigenfunction associated to ˆλ 1. We are going to prove that for K 1 and K 2 large enough, the above couple is sub-supersolution of (3.1).

13 Age-dependent Lotka-Volterra diffusive systems 13 Indeed, it is clear that u, v > on Ω, and u(a, x, ) > u and v(a, x, ) > v for large K 1 and K 2. On the other hand, u(, x, t) = K 1 ˆϕ 1 = K 1 ˆϕ 1 Bg(a)da β 1 (x, a, t)k 1 g(a) ˆϕ 1 da = β 1 (x, a, t)u(x, a, t)da. Finally, it is not hard to prove that in the equations we need verify the following conditions g(a) ˆϕ 1 (K 1 bk 2 ) λ ˆλ 1 + r m + m(a) µ 1, g(a) ˆϕ 1 (K 2 ck 1 ) ν ˆλ 1 + r m + m(a) µ 2. Since bc < 1, we can choose K 1 and K 2 sufficiently large so that the above inequalities hold. Take now a bounded solution (u, v). Then, there exist K 1 and K 2 sufficiently large such that u K 1 g(a) ˆϕ 1 and v K 2 g(a) ˆϕ 1 and (K 1 g(a) ˆϕ 1, K 2 g(a) ˆϕ 1 ) supersolution of (3.1). This concludes the proof in a similar way as before. 4 Blow-up in finite time: strong cooperating case In the rest of the paper, our aim is to prove that in the cooperating case, when bc > 1 then the solution blows up in finite time. In order to prove the result, we need some previous ones and some notations. In [1], it was proved the existence of a principal eigenvalue, denoted by λ, of the problem u a u + µ(x, a)u = λu in := Ω (, A ), u(x, a) = on Σ := Ω (, A ), (4.1) where u(x, ) = β(x, a)u(x, a)da in Ω, (H µ ) µ is a function such that µ L (Ω (, r)) for r < A and r µ M (a)da <, µ L (a)da = +, (4.2) being µ L (a) := ess inf x Ω µ(x, a) and µ M (a) := ess sup x Ω µ(x, a).

14 14 M. Delgado and A. Suárez (H β ) β L (), β, nontrivial and meas{a [, A ] : β L (a) := ess inf x Ω β(x, a) > } >. We need recall the some points of the proof of this result. For each φ L 2 (Ω) we define z φ the unique solution of z a z + µ(x, a)z = in, z = on Σ, z(x, ) = φ(x) in Ω, (4.3) and define the operator B λ : L 2 (Ω) L 2 (Ω) by B λ (φ) = β(x, a)e λa z φ (x, a)da. The operator B λ is positive and compact. Denoting by r(b λ ) its spectral radius, we prove in [1] that there exists a unique value of λ, λ such that r(b λ ) = 1. So, by the Krein- Rutman s Theorem, there exists a positive function φ > such that B λ φ = φ. It is not difficult to prove that ϕ := e λa z φ is the eigenfunction associated to λ. Again, by the Krein-Rutman s Theorem, if we denote as B λ B λ, r(bλ ) = 1, and so there exists ψ > such that the adjoint operator of B λ ψ = ψ. We calculate heuristically B λ. For each ψ L 2 (Ω), denote by v ψ the unique solution (see Lemma 4.1 below) of the backward problem v a v + µ(x, a)v = β(x, a)e λ a ψ(x) in, v = on Σ, (4.4) v(x, A ) = in Ω. Let φ, ψ L 2 (Ω). Then, B λ (φ), ψ = [ β(x, a)e λa z φ (x, a)da]ψ(x)dx = Ω β(x, a)e λ a z φ (x, a)ψ(x)dadx =

15 Age-dependent Lotka-Volterra diffusive systems 15 ( (v ψ ) a v ψ + µv ψ )z φ (x, a)dadx = [ v ψ (x, A )z φ (x, A ) + v ψ (x, )z φ (x, )]dx+ ((z φ ) a z φ + µz φ )v ψ dadx = Ω whence Bλ (ψ) = v ψ (x, ). Taking now Ω φ(x)v ψ (x, )dx = φ, B λ (ψ), it is easy to prove that ϕ verifies ϕ := e λ a v ψ w a w + µ(x, a)w = λ w + β(x, a)w(x, ) in, w(x, a) = on Σ, (4.5) w(x, A ) = in Ω. We formalize this calculation. Lemma 4.1 Let ψ L 2 (Ω) and assume (H5) There exists A < A such that supp(β) Ω (, A ). There exists a unique solution v L 2 () of the backward problem (4.4). Moreover, v L 2 (, A ; H 1 (Ω)) and there exists the value v(x, ) L2 (Ω). Remark 4.2 Hypothesis (H5) has been considered previously by many authors, [4], [5]. It has an obvious biological sense: the temporary component of the fertility rate is contained in (, A ), i.e., in a neighborhood of A the species has not got reproductive capacity. Proof: Denoting by µ(x, s) := µ(x, A s) and β(x, s) := β(x, A s), system (4.4) is equivalent to w s w + µ(x, s)w = β(x, s)e λ (A s) ψ(x) in, w = on Σ, (4.6) w(x, ) = in Ω, and now we are interested in the value w(x, A ), with the change of variable w(x, s) = v(x, A s).

16 16 M. Delgado and A. Suárez Under the change of variable z = e ks w, k >, z satisfies z s z + ( µ + k)z = g(x, s) := β(x, s)e λ (A s) ks ψ(x) in, z = on Σ, (4.7) z(x, ) = in Ω, and so by (H µ ), we can take k large such that µ + k/3. We study now (4.7) instead of (4.6). Define µ n := min{ µ, n}, n IN, and consider the problem z s z + ( µ n (x, s) + k)z = g(x, s) in, z = on Σ, (4.8) z(x, ) = in Ω. Now, for each n IN, since µ n + k is bounded, there exists a unique z n solution of (4.8) with z n C([, A ]; H 1 (Ω)) L 2 (, A ; H 2 (Ω) H 1 (Ω)), (z n ) s L 2 (, A ; L 2 (Ω)). Multiplying (4.8) by z n and integrating we obtain 1 d z n 2 + z n 2 + ( µ n + k)zn 2 = gz n, 2 ds and so, applying that 2ab (ε 2 a 2 + (1/ε 2 )b 2 ) we get 1 d z n 2 + z n 2 + ( µ n + k/3)zn 2 + (k/3)zn 2 C. 2 ds Now, we can extract a sequence (z n ) such that z n z µn + (k/3)z n h in L 2 (, A ; H 1 (Ω)), in L 2 (), (z n ) s + ( µ n + k/3)z n j in L 2 (, A ; H 1 (Ω)).

17 Age-dependent Lotka-Volterra diffusive systems 17 On the other hand, for ϕ C c (, A ; H 1 (Ω)), and for n large enough, we get (z n ) s + ( µ n + k/3)z n, ϕ = ( z n ϕ s + ( µ + k/3)z n ϕ) and so ( zϕ s + ( µ + k/3)zϕ) (z s + ( µ + k/3)z)ϕ, j = z s + ( µ + k/3)z. Similarly, it can be proved that h = µ + k/3z. This shows that z is solution of (4.7). For the uniqueness, take two different solutions w 1 and w 2 of (4.6). Then, w = w 1 w 2 satisfies that w s w + µ(x, s)w =, in, w = on Σ, w(x, ) = in Ω. It suffices to multiply this problem by w and obtain that w. Now, define µ L (s) := µ L (A s), µ M (s) := µ M (A s) according to (4.2). By the maximum principle, if w is solution of (4.6), then w M w w L, (4.9) where w M and w L are the respective solutions of (4.6) with µ = µ M and µ L. We study now (4.6) with µ = µ M ; a similar study could be made with µ L. perform the change of variable If we s z = exp ( µ M (s)ds)w, (4.6) transforms into z s z = h(x, s) := β(x, s)exp ( s µ M(σ)dσ)e λ (A s) ψ(x) in, z = on Σ, (4.1) z(x, ) = in Ω. But, thanks to (H5) h(x, s) L 2 (),

18 18 M. Delgado and A. Suárez and so w M is well-defined and w M C([, A ]; L 2 (Ω)). This shows (4.9) and an application of convergence dominated Theorem proves that w(x, A ) := lim a A w(x, a) is well-defined. In order to prove the blow-up result, we need more regularity of the solution on the variable t. For that, we will use semigroup theory. Specifically, define X := L 2 () and the operator A : X X as ψ(x, a) Aψ := µ(x, a)ψ(x, a) + ψ(x, a), { a D(A) := ψ X : Aψ X, ψ Ω =, ψ(x, ) = ψ D(A), with β(x, a)ψ(x, a)da In [12] it was proved, Theorem 1, that A is the infinitesimal generator of a C -semigroup on the state space X. Consider the equation u t + u a u + µ(x, a)u = u(γ + u) in Ω O, }. u(x, a, t) = on Ω O, u(x, a, ) = u (x, a) in Ω (, A ), u(x,, t) = β(x, a)u(x, a, t)da in Ω (, T ). (4.11) Proposition 4.3 Assume (H µ ), (H β ), (H5), γ > λ and u D(A). Then, there exists a unique solution of (4.11) in (, T ) for some T >. Moreover, the positive solution of (4.11) blows up in finite time. Proof: Observe that (4.11) can be written as an evolutionary equation on X u t = Au + F (u), u(x, a, ) = u (x, a), (4.12) where F (u) = u(γ + u). Since, F is locally Lipschitz, it follows the existence of a local solution in C 1 ([, T ]; X), see for instance [7]. Now, let u a positive solution of (4.11), and consider q(t) := u(x, a, t)ϕ (x, a)dadx.

19 Age-dependent Lotka-Volterra diffusive systems 19 Then, recalling that ϕ verifies (4.5), we get q (t) = u t ϕ dadx = ( u a + u µu + γu + u 2 )ϕ dadx = Ω [ u(x, A )ϕ (x, A ) + u(x, )ϕ (x, )]dx + γq + [β(x, a)ϕ (x, ) µϕ + (ϕ ) a + ϕ ]u + γq + On the other hand, by the Holder inequality and so, ϕ u = whence the blow-up follows. ((ϕ ) a + ϕ µϕ )u + u 2 ϕ = u 2 ϕ = (γ λ )q + u 2 ϕ. u(ϕ ) 1/2 (ϕ ) 1/2 [ u 2 ϕ ] 1/2 [ ϕ ] 1/2, q (t) (γ λ )q + C 1 q 2 (t), q() = u (x, a)ϕ (x, a)dadx := q >, We can prove now blow-up result for the strong cooperative case. Theorem 4.4 Assume b, c > and bc > 1. Let µ i, β i i = 1, 2 satisfying (H µ ), (H β ) and (H5). Take u, v D(A) positive and λ, ν > λ. Then, the solutions of (3.1) blow-up in finite time. Proof: Take λ, ν > λ. Consider K 1 := 1 + b bc 1, K 2 := 1 + c bc 1, γ := min{λ, ν} > λ, µ(x, a) := max{µ 1, µ 2 }, β(x, a) := min{β 1, β 2 } and w := min{ 1 K 1 u, 1 K 2 v }. Denote by w γ the solution of (4.11) with the above data. Then, it is not difficult to prove that is a sub-solution of (3.1) provided of (u, v) = (K 1 w γ, K 2 w γ ) bk 2 K 1 1 and ck 1 K 2 1, which is true by the definitions of K 1 and K 2. This concludes the result. Acknowledgements: A.S. thanks to M. Molina-Becerra for helpful comments which have improved the paper.

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