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Spectra of structured diffusive population equations with generalized Wentzell-Robin boundary conditions and related topics

This paper provides two different extensions of a previous joint work "Time asymptotics of structured populations with diffusion and dynamic boundary conditions; Discrete Cont Dyn Syst, Series B, 23 (10) (2018)" devoted to asynchronous exponential asymptotics for bounded and weakly compact reproduction operators. The first extension considers bounded non weakly compact reproduction operators while the second extension deals with unbounded kernel reproduction operators and needs, as a preliminary step, a new generation result.

with m < +∞ and b 0 − γ(0) > 0, b m + γ(m) > 0 (and initial conditions) was considered first in [5] and then in [10]. We refer the reader to the introductions of these two papers and to the references therein for more motivation and information on such structured population models. From a mathematical point of view, the main issues are the well-posedness of this evolution system and the understanding of its time asymptotics. The goal of the present paper is to extend [10] to much more general reproduction operators.
In [5][10], the Cauchy problem above, for m < +∞, is written in the following matrix form in the space The domain of A is and A is shown to be the generator of a positive C 0 -semigroup (T (t)) t≥0 . Moreover, by using the Hopf maximum principle, (T (t)) t≥0 is shown to be irreducible, or equivalently the resolvent of its generator is positivity improving, see ([10] Theorem 2.4 and Proposition 2). Since the perturbing operator K is bounded then is a generator of a positive C 0 -semigroup (U (t)) t≥0 . The fact that U (t) ≥ T (t) implies trivially that (U (t)) t≥0 is also irreducible regardless of K.
Note that the boundedness of K amounts to and Besides the generation theory, various mathematical results are given in [10]; in particular, the two semigroups have the same essential spectrum σ ess (U (t)) = σ ess (T (t)), (and consequently the same essential type ω ess (T ) = ω ess (U )) under the assumption that the reproduction operator from which we can derive that (U (t)) t≥0 has a spectral gap and exhibits an asynchronous exponential behaviour, see ([10] Theorem 3.2 and Theorem 2.9). The object of the present paper is to give two different extensions of the general theory given in [10].
1. The first extension consists in replacing (14) by the much weaker conditions: (i) β 0 + β m is not identically zero and the reproduction operator β is an arbitrary positive bounded operator or (ii) β 0 = β m = 0 and the reproduction operator β is a positive bounded operator which dominates a positive non trivial weakly compact operator β.
(Note that in both cases, β need not be a kernel operator.) 2. Another extension consists in dealing with a class of unbounded kernel reproduction operators β : ϕ → m 0 β(s, y)ϕ(y)dy (0 < m ≤ +∞), and unbounded functions β 0 , β m . The strategy in the first extension consists in considering A as a (non trivial) weakly compact perturbation of A : in the case (ii). It turns out that this slight change of point of view enlarges considerably the class of bounded positive reproduction operators β for which the general theory in [10] is still valid, even if we don't know the possible biological significance of non kernel reproduction operators.
Our second extension of [10] is much more involved. Firstly, we are faced with well posedness of the Cauchy problem. Our generation theory relies on Desh's theorem [3] [15] via suitable weak compactness arguments [9]. Secondly, the proof of a stability of essential spectra turns out to be much more technical than in [10].
Our paper is organized as follows.
In Section 2, we provide some important functional analytic reminders; in particular, we recall a generation theorem for unbounded perturbations in L 1 spaces, see Lemma 2.3. In Section 3, we extend the asynchronous exponential behaviours (for m < +∞ or m = +∞) given in [10] when β 0 + β m is not identically zero and the reproduction operator β is an arbitrary positive bounded operator or when β 0 = β m = 0 and the reproduction operator β is a positive bounded operator which dominates a non trivial positive weakly compact operator, see Theorem 3.1 and Theorem 3.2. The condition that β dominates a non trivial positive weakly compact operator is satisfied e.g. if β is a kernel operator whose kernel β(., .) is such that there exists some m ∈ [0, m) such that β(., .) is not identically zero in a neighborhood of (m, m) and there exists α ∈ [0, 1) such that lim sup We show that K is A-weakly compact so that, by Lemma 2.3, A := A + K is a generator of a positive C 0 -semigroup (U (t)) t≥0 , see Theorem 4.1. By using a suitable approximation procedure, we show that U (t) − T (t) is weakly compact and consequently (U (t)) t≥0 and (T (t)) t≥0 have the same essential spectrum and the same essential type, see Theorem 4.2. Finally, the proofs of the asynchronous exponential asymptotics for m < +∞ or m = +∞ (Theorem 4.3 and Theorem 4.4) follow the same strategy as for bounded perturbations. Notice that a further (more abstract) extension of [10] is still possible, see Remark 3. This paper is dedicated to the celebration of the 60th birthday of Gisèle Ruiz Goldstein.
2. Some functional analytic reminders. Let X be a Banach lattice. We start with: where s(A) and s(T ) denote respectively the spectral bounds of A and T .
We recall that the spectral bound of a closed (unbounded) operator O is defined by Ox .

It follows easily that if
. We complement this by: We give now a generation result for unbounded perturbations. We recall also that the essential type ω ess (S) of a C 0 -semigroup (S(t)) t≥0 satisfies r ess (S(t)) = e ωess(S)t (t ≥ 0) where r ess (S(t)) = sup {|λ| ; λ ∈ σ ess (S(t))} see e.g. [11]. Finally, we recall that, for a positive C 0 -semigroup (S(t)) t≥0 on L p spaces, the type coincides with the spectral bound of its generator, see e.g. [17].
3. Bounded reproduction operators. We start with a sufficient condition for a bounded kernel operator to dominate a weakly compact operator.
Then β dominates a non trivial weakly compact operator.
Proof. We note that, in the statement above, m = m is allowed if m < +∞. We observe that (16) implies the existence of a bounded interval I ⊂ [0, m) such that |s − y| α β(s, y) is bounded (and not identically zero) on I × I β(s, y) ≤ C |s − y| α , (s, y) ∈ I × I.
It is easy to see that the integral operator L 1 (I) → L 1 (I) with kernel is a non trivial positive weakly compact operator dominated by β.
Remark 1. We suspect that the assumption that β dominates a non trivial weakly compact operator has to do with the spectral condition r ess (β) < r σ (β) but we have not yet a complete proof.
3.1. Asynchronous exponential behaviour. We already know that (T (t)) t≥0 and (U (t)) t≥0 are irreducible ([10] Theorem 2.4 and Proposition 2). We start with the case m < +∞. and there exists ε > 0 and C > 0 such that where P is the rank one spectral projection associated to the leading eigenvalue s(A).
Proof. It suffices to show that ω ess (U ) < ω(U ) and to invok a classical functional analytic result, e.g. Consider first the case: (i ) β 0 or β m is not identically zero.
Let T (t) t≥0 be the semigroup generated by A. We have the Duhamel equation is a compact operator (see e.g. [16]). This implies that (U (t)) t≥0 and T (t) t≥0 have the same essential spectrum [6] and consequently the same essential type ω ess (U ) = ω ess ( T ).
We note that the irreducible compact operators (λ − A) −1 and (λ − A) −1 are such that Consider now the case: (ii) β 0 = β m = 0 and the reproduction operator β dominates a positive non trivial weakly compact operator β. Let now Let T (t) t≥0 be the semigroup generated by A. Let β(., .) be the kernel of β, (note that a weakly compact operator is a kernel operator, see [4], p. 508). As previously, we have a Duhamel equation is a weakly compact operator (see [14] or [8]). This implies that (U (t)) t≥0 and have also the same essential spectrum [6] and consequently the same essential type ω ess (U ) = ω ess ( T ).
Arguing as previously, one sees that s( A) < s(A) (since K = 0) and which ends the proof.
We consider now the case m = +∞.
Theorem 3.2. Let m = +∞ and let Assumptions (6)(7)(10) be satisfied. We assume that the reproduction operator β is a positive bounded operator. We assume that either (i) β 0 + β m is not identically zero or (ii) that β 0 = β m = 0 and the reproduction operator β dominates a positive non trivial weakly compact operator β. Let A := A + K where K is given by (17) in the case (i) or by (18) then ω ess (U ) < ω(U ) = s(A) and there exists ε > 0 and C > 0 such that where P is the rank one spectral projection associated to the leading eigenvalue s(A).
Proof. By combining (19) and the characterization of the perturbed spectral bound (15) we get

s( A) < s(A).
As previously, (U (t)) t≥0 and T (t) Remark 2. We recall that (U (t)) t≥0 and (T (t)) t≥0 satisfy the Duhamel equation see [13]. In particular extends uniquely as a bounded operator on X . We will denote this extension symbolically by t 0 U (t − s)KT (s)ds. 4.2. Stability of essential spectra. Our second key result is: Let m < +∞ and Assumptions (1)(5) be satisfied or let m = +∞ and Assumptions (6)(10) be satisfied. Then U (t) − T (t) is a weakly compact operator and therefore (U (t)) t≥0 and (T (t)) t≥0 have the same essential spectrum and consequently the same essential type.
Note that T (s)ϕ ∈ D(A) so KT (s)ϕ is meaningfull since K is A-bounded. There exists C ≥ 1 and λ ∈ R such that U (t) ≤ Ce λt . Then Since K − K k is positive then, by the additivity of the L 1 norm on the positive cone, We note that for any constant c > 0 and consequently in operator norm. Since t 0 U (t−s)K k T (s)ds (k ∈ N) are weakly compact operators then so is t 0 U (t−s)KT (s)ds. Finally, for any t > 0, U (t)−T (t) is a weakly compact operator and therefore (U (t)) t≥0 and (T (t)) t≥0 have the same essential spectrum, see [6].

4.3.
Asynchronous exponential behaviour. We know that A is a generator of a positive semigroup (U (t)) t≥0 (Theorem 4.1) Since U (t) ≥ T (t) then (U (t)) t≥0 is also irreducible. Since (U (t)) t≥0 and (T (t)) t≥0 have the same essential type (Theorem 4.2) then we have all the ingredients to deduce the following results whose proofs are similar to those of Theorem 3.1 and Theorem 3.2 and are omitted.
where P is the rank one spectral projection associated to the leading eigenvalue s(A). where P is the rank one spectral projection associated to the leading eigenvalue s(A).
Remark 3. In the same spirit, we could combine the two extensions given in this paper into one (more general) abstract construction. For instance, for m < +∞, we may replace (11) by  In this case A := A + K is a generator of a positive C 0 -semigroup (U (t)) t≥0 on X , see [3] [15]. We can recover the asynchronous exponential behaviour of (U (t)) t≥0 once (21) is not zero. If (21) is zero, we assume that β splits as β = β 1 +β 2 where β 1 is (say) a positive bounded operator L 1 (0, m) → L 1 (0, m)