A singular perturbation result for a class of periodic-parabolic BVPs

: In this article, we obtain a very sharp version of some singular perturbation results going back to Dancer and Hess [ Behaviour of a semilinear periodic-parabolic problem when a parameter is small , Lecture Notes in Mathematics, Vol. 1450, Springer-Verlag, Berlin, 1990, pp. 12 – 19] and Daners and López-Gómez [ The singular perturbation problem for the periodic-parabolic logistic equation with inde ﬁ nite weight functions , J. Dynam. Di ﬀ erential Equations 6 (1994), 659 – 670] valid for a general class of semilinear periodic-parabolic problems of logistic type under general boundary conditions of mixed type. The results of Dancer and Hess [ Behaviour of a semilinear periodic-parabolic problem when a parameter is small , Lecture Notes in Mathematics, Vol. 1450, Springer-Verlag, Berlin, 1990, pp. 12 – 19] and [ The singular perturbation problem for the periodic-parabolic logistic equation with inde ﬁ nite weight functions , J. Dynam. Di ﬀ erential Equations 6 (1994), 659 – 670] were found, respectively, for Neumann and Dirichlet boundary conditions with L = − Δ . In this article, L stands for a general second-order elliptic operator.


Introduction
In this article, we study the periodic-parabolic problem where > p 1 and > d 0 are the constants, under the following conditions: (i) Ω is a bounded domain of N , ≥ N 1, of class +θ 2 for some ( ) , where Γ 0 and Γ 1 are two disjoint open and closed subsets of ∂Ω.As they are disjoint, Γ 0 and Γ 1 are of class +θ 2 .(ii) For a given > T 0, L stands for the autonomous linear second-order differential operator for every 1 , where ( ) is an outward pointing nowhere tangent vector field.
As it will become apparent in Section 3, though in this article, ( ) β x can change of sign on Γ 1 , one can assume, without loss of generality, that Note that since Γ 1 is smooth, it must consist of finitely many components, say Γ j 1, with { } ∈ j q 1, …, for some integer ≥ q 1.Throughout this article, for every continuous T -periodic function [ ] → V T : 0, , we will denote by , and let ⊂ ∪ K Ω Γ 1 be a compact subset.Then, the following conditions are satisfied: where [ ] α m a , stands for the unique positive periodic solution of the associated kinetic model (iii) If ( ) > m x 0 for all ∈ x K and there exists a nonempty subset I { } ⊂ q 1, …, such that This result is a substantial extension of some not well-known findings of Dancer and Hess [5], and Theorem 1.3 of Daners and López-Gómez [6], which are very simple counterparts of Theorem 1.1 for L = −Δ and either ∂ = Ω Γ 1 with = β 0, or ∂ = Ω Γ 0 , respectively.Some very recent elliptic counterparts of Theorem 1.1 valid for general elliptic operators L B ( ) , , Ω have been given by Fernández-Rincón and López-Gómez [7].In this article, it remains an open problem to ascertain whether, or not, the condition that ( ) ( ( ) ( )) = m a m t a t , , on a neighborhood of ∂ ∩ K Γ 1 in Part (iii) is of a technical nature.Theorem 1.1 is of a huge interest in population dynamics, where the behavior of the species for small diffusion coefficients provides us with an idealized behavior of many real systems.A simple glance to the pioneering article of Hutson et al. [9] will convince the reader of it very easily.Actually, [9] generated a huge industry in the field under the auspices of Y. Lou, W. M. Ni, and their coworkers.The reader should compare the results of Section 2 of Hutson et al. [9] with Theorem 1.1 of Lou [13].
The condition ( ) > a x t , 0for all ( ) ∈ × x t , Ω ¯is imperative for the existence of a positive solution of (1.1) for small > d 0 even for the simplest elliptic counterpart of (1.1) where > m 0 is a constant.Indeed, if ( ) a x vanishes on some nice smooth subdomain of Ω, say Ω 0 , with ⊂ Ω ¯Ω 0 , then, according to [12,Ch. 4], it is well-known that (1.5) possesses a positive solution if, and only if, where we are denoting D B ≡ if = ∅ Γ 1 .Since (1.6) can be equivalently expressed as it is apparent that (1.5) cannot admit a positive solution for sufficiently small > d 0.
Naturally, we are denoting Also, for every > d 0, we will consider the periodic-para- bolic operator and for any subdomain ⊂ D Ω, we denote by T .This study is organized as follows.In Section 2, we analyze the associated kinetic problem (1.4).In Section 3, we show that, without loss of generality, one can assume that (1.2) holds in (1.1).In Section 4, we study some pivotal properties of the underlying principal eigenvalues associated with the periodic-parabolic problem (1.1).In Section 5, we study the existence and the uniqueness of the positive solution of (1.1) for small > d 0. In Section 6 we construct some supersolutions for problem (1.1).In Section 7, we construct some subsolutions of (1.1) in the special case when ( ) . The construction of φ in the proof of Theorem 7.1 is a technical device borrowed from López-Gómez [10].In Section 8, we deliver an auxiliary result to prove Theorem 1.1(iii).Finally, in Section 9, the proof of Theorem 1.1 is completed.This section analyzes the existence of (T -periodic) positive solutions of where ∈ x Ω ¯is regarded as a parameter.Its main result can be stated as follows.
In such case, it is unique and given through where .
is the unique non-negative T-periodic solution of (2.1).
Proof.Since > p 1, for every ∈ x Ω ¯, any solution of (2.1) satisfies and hence, u t is a positive solution of (2.1) for some ∈ x Ω ¯.Then, the change of variable = − v u p 1 transforms (2.1) into the linear problem Solving the linear differential equation of (2.4), we have that , the following identity must be satisfied: Consequently, substituting (2.6) into (2.5) and taking into account that = − u v p  (2.6).As, in particular, ( ) > v 0 0, necessarily, ( ) > m x 0. Finally, the uniqueness of the T -periodic positive solution comes from the fact that it must be given by (2.3).This ends the proof.□ Subsequently, we denote by [ ] α m a , the function where The next result collects some of its properties.
, .Thus, it satisfies the following properties: , then there exists a neighborhood, , of x 0 in Ω ¯such that ( ) (2.9) Proof.By the assumptions, A singular perturbation result for a class of periodic-parabolic BVPs  5 Thus, thanks to Proposition 2.1, m a x m a T , ; , Unfortunately, Estimate (2.9) cannot be obtained directly from (2.3), because the character of the integral is unclear when a decreases and m increases.Thus, to prove (2.9), in this case, we use the following argument. Setting we have that In other words, setting Then, setting for some constant > μ 0 to be determined later, we have that Thus, making the change of variable where where L h stands for the differential operator A singular perturbation result for a class of periodic-parabolic BVPs  7 The reader is sent to Section 1.7 of [11] for any further details on the change (3.1), going back to the generalized maximum principle of Protter and Weinberger [14].Note that (3.3) satisfies similar properties as L.
In particular, its coefficients also belong to F .Since ( ) h x does not depend on t, by (3.2), the change of variable (3.1) transforms the periodic-parabolic equation of (1.1) into , w h e r e , , .
Thus, since where we are denoting we have that Hence, choosing μ to satisfy (3.5), we have that ( ) > β x 0 h for all ∈ x Γ 1 .Summarizing, for sufficiently large μ, the change of variable (3.1) transforms problem (1.1) into the next one where L h and B h are given by (3.3) and (3.4) with β h satisfying (3.6).As the regularity of the several coefficients involved in the framework of (3.7) is the same as those imposed in (1.1), in this article, we will work with problem (1.1) assuming, without loss of generality, that condition (1.2) holds.
Suppose, in addition, that ( ) , for all ∈ x Ω ¯, and that , is the unique positive solution of (2.1).Then, thanks to Proposition 2.1, and enlarging μ so that, instead of (3.5), the next (strongest) condition holds then, besides (3.6), one can also obtain that Indeed, along Γ 1 , one has that Thus, as soon as μ satisfies (3.8), we have that Therefore, (3.8) implies (3.9 Consequently, throughout this article, besides condition (1.2), we can assume, without loss of generality, that when

Auxiliary eigenvalue problem
In this section, we focus our attention into the eigenvalue problem Thanks to Hess [8] and Antón and López-Gómez [2,3], [4,Sec. 6], problem (4.1) possesses a unique principal eigenvalue, denoted by P , which is algebraically simple and strictly dominant.To state its main monotonicity properties, we need to introduce some notation.Subsequently, for any proper subdomain we will denote by B Ω0 the boundary operator defined by The next result goes back to Antón and López-Gómez [4,Sec. 7].It collects some important monotonicity properties of that will be used throughout this article.Proposition 4.1.Under the general assumptions of this article, the following properties are satisfied: T Ω ¯0, .Then, A singular perturbation result for a class of periodic-parabolic BVPs  9 (ii) Let Ω 0 be a proper subdomain of Ω of class +θ 2 satisfying (4.2), and ∈ V F .Then, The main result of this section reads as follows.It ascertains the value of P and finds from it its asymptotic behavior as ↓ d 0 when ( ) Then, the principal eigenvalue of the problem is given through where stands for the principal eigenvalue of the linear elliptic eigenvalue problem Moreover, up to a positive multiplicative constant, the principal eigenfunction ( ) can be expressed through where ( ) . Thus, Proof.The existence and the uniqueness of ( ) λ ψ , 1, is a direct consequence of Antón and López-Gómez [3,4].To prove the theorem, we will search for a T -periodic positive function, ( ) γ t , for which provides us with a principal eigenfunction of (4.3).By the choice of φ 1 , Moreover, inserting ψ 1 into the differential equation of (4.3), we are driven to which can be equivalently expressed as Thus, and hence, Since ( ) γ t is T -periodic and positive, we have that ( ) ( ) = > γ T γ 0 0, and therefore, Consequently, by uniqueness, provides us with the principal eigenvalue of (4.
In such case, the positive solution is unique; throughout this article, it will be denoted by [ ] θ m a d , , , and the following holds: Next result gives some comparison results that will be used later.
Proposition 5.1.Under condition (5.1), the following properties are satisfied: (i) For every subsolution, ⪈ u 0, of (1.1), one has that (5.2) Proof.By the uniqueness of the positive solution, if u is not a strict subsolution of (1.1), i.e., if u solves (1.1).Thus, we will assume that ⪈ u 0 is a strict subsolution of (1.1).Then, [ ] we have that A singular perturbation result for a class of periodic-parabolic BVPs  11 with some of these inequalities strict.On the other hand, since ⪈ u 0, we find that Thus, it follows from Proposition 4.1(i) that Hence, thanks to Theorem 1.1 of Antón and López-Gómez [3], we find from (5.3) that which ends the proof of Part (i).The proof of Part (ii) follows the same general patterns as the proof of Part (i).Thus, we will omit its technical details here.We now prove Part (iii).Since is a subsolution of . Thus, (5.2) follows from Part (i).This ends the proof.□ The following result gives a sufficient condition for the existence of positive solutions of (1.1) for small diffusions.Proof.Thanks to (5.4), ( ) . By the uniform continuity of ( ) , f o r a l l , ¯0, .
because of the choice of ε.
The following result gives a sufficient condition for the nonexistence of positive solution of (1.1) for small diffusions.

11)
A singular perturbation result for a class of periodic-parabolic BVPs  13 Moreover, by Theorem 4.1, it follows from (5.9) that (5.12) As (5.10) contradicts (5.11) and (5.12), the proof is complete.□ 6 Constructing supersolutions Proposition 6.1.Assume Then, for each > ε 0, there exists Then, it follows from (6.4) that Subsequently, we will prove that u ¯δ is a positive strict supersolution of (1.1).Indeed, in Ω T , we find that Note that L because we are assuming that a and m are of class 2 in ∈ x Ω ¯.Thus, thanks to (6.3), it becomes apparent from (6.6) that there exists Thanks to (6.9), we have that Thus, by (6.7), (6.8), and (6.10), u ¯δ is a positive strict supersolution of (1.1).Therefore, thanks to Proposition 5.1(ii), (6.5) implies that The first estimate of (6.12) follows from (6.11) and Proposition 2.3.The second estimate holds true from the continuity of [ ] α m a , with respect to m and a, which is a direct consequence from (2.5) and (2.6).Then, by (6.1), it follows from (6.11) that Thus, by the previous case, there exists ( ) (6.13) Moreover, due to Proposition 5.1(iii), it follows from (6.11) that Then, thanks to (6.12), (6.13), and (6.14), we find that, for every This shows (6.2) and ends the proof.□ 7 Constructing subsolutions for m and a autonomous in ∈ ∈ x Ω Theorem 7.1.Assume that with Let ⊂ ∪ K Ω Γ 1 be a compact set.Then, for every > ε 0, there exists ( ) > d ε K , 0 such that, for every A singular perturbation result for a class of periodic-parabolic BVPs  15 Proof.Pick > ε 0. The proof will be distributed into four steps.Thanks to (7.2), it follows from Proposition 2.2(iii) that Step 1: In this step, we are going to prove that, for every ∈ x Ω 0 , there exist , for all , 0, and 0, ¯. .Now, let us consider the ρ-neighborhood of ∂B in B, as well as a function of the type where ξ is any regular extension of | φ 1 ¯ρ to B such that 1, in \ , max 1, 0 1, for all ¯\ .
ρ B ¯0 0 Finally, for every > δ 0, we consider the function We are going to show that, for every ( ) ∈ δ 0, 1 and sufficiently small > d 0, u δ provides us with a positive strict subsolution of the problem , it follows from the definition of ( ) [ ] α t m a , and (7.5) that in B\ ρ , we find that, in B\ ρ , , , , there is such that, for every Thus, setting it follows from (7.8) and (7.9) that, for every on ∂B, we also have that .Thus, since u δ is a positive strict subsolution of (7.7) for each ( ) it follows from Proposition 5.1(i) applied in B that, for every On the other hand, by construction, there exist , for all , 0, .
, it suffices to take a sufficiently small > R 0 1 and δ* sufficiently close to 1.For these choices, it follows from (7.13) that , for all , 0, .
Step 2: In this step, we will prove (7.3) in the particular case when ⊂ K Ω.In such case, according to Step 1, for every ∈ x K, there exist By the compactness of K , there are an integer ≥ m 1 and m points ∈ it follows from (7.17) that, for every As this provides us with (7.3), the proof of the theorem is completed if In this step, we consider one of these components, say Γ i 1, , and, for every By construction, ˜Γ ˜Γ , and ˜Γ ˜Γ .
This step shows that there exist and Indeed, for sufficiently small > R 0, let us denote by 1, the principal eigenpair of the linear eigenvalue problem By construction, for sufficiently small > ρ 0, we have that Next, we will fix one of those ρ's and consider the function where η i is a regular extension of φ ˜i 1, from and , for all Γ .
In the same vein as in Step 1, for every > δ 0, we consider the function Similarly, we will show that, for every ( ) ∈ δ 0, 1 and sufficiently small > d 0, the function u ˜δ i , is a positive subsolution of the problem , it follows from (7.20) that, in the annular cylinder , one has that Thus, since for all ∈ x ˜R i , , it follows from (7.24) that there exists such that, for every Similarly, since > p 1 and < < δ 0 1, it follows from (7.21) that and hence, A singular perturbation result for a class of periodic-parabolic BVPs  19 Thus, choosing we find from (7.25) and (7.27) that, for every As to the boundary conditions concerns, by construction, we have that ˜0, for all , Γ ˜0, .
Moreover, thanks to (7.22), on , we find that On the other hand, by construction, it follows from (7.22) that there exist ( ) ( ) Moreover, as the unique positive solution [ ] θ m a d , , of (1.1) is a positive strict supersolution of (7.23), it follows from Proposition 5.1 that, for every Finally, combining (7.34) with (7.35), (7.19) holds.This ends the proof of Step 3.
Step 4: Finally, we are going to prove (7.3) for every compact subset K of ∪ Ω Γ 1 .First, we consider Then, thanks to Step 3, for every Now, let K be a compact subset of ∪ Ω Γ 1 .Then, . Hence, in Ω T , we have that A singular perturbation result for a class of periodic-parabolic BVPs  21 Proof.Thanks to Proposition 6.1, for any given > ε 0, there exists Finally, combining (8.7) with (8.10) and (8.9), Estimate (8.4) holds, and the proof of Step 1 is complete.
Step 2: In this step, we will prove that, for every compact subset ⊂ K Ω and > ε 0, there exists Indeed, since where ( ) R x is the radius associated with ∈ x K constructed in Step 1, by the compactness of K , there is a finite subset of K , say { Fix > ε 0.Then, thanks to Step 1, we already know that, for every Therefore, setting it follows from (8.12) and (8.13) that (8.11) holds.This ends the proof of Step 2.
Step 4: Since K is a compact subset of ∪ Ω Γ 1 , we have that where K Γ1 is the compact set defined in (8.14).
A singular perturbation result for a class of periodic-parabolic BVPs  23 Applying (8.11) in the compact set , there exists , for all 0,min ˜, ˆ.

Main result
The main result of this article reads as follows.
Theorem 9.1.Assume that there exists , and let ⊂ ∪ K Ω Γ 1 be a compact subset.Then, the following conditions are satisfied: (ii) If ( ) > m x 0, for all ∈ x K and ⊂ K Ω, then (iii) If ( ) > m x 0 for all ∈ x K and there exists a nonempty subset I { } ⊂ q 1, …, such that Our main goal in this article is to obtain the following singular perturbation result, where [ ] θ m a d , , stands for the unique positive solution of the semilinear periodic-parabolic problem (1.1).According to Theorem 5.2, [ ] θ m a d , , exists for sufficiently small > d 0 if ( )

Theorem 5 . 2 . 5 )
If there exists ∈ x Thus, thanks to Theorem 5.1, (1.1) has a unique positive solution for all ( ] , u δ provides us with a positive strict subsolution of (7.7).Thanks to (7.2) and applying Theorem 5, problem (7.7) has a unique positive solution, denoted by [ ] θ m a d B , , ;

1 ( 7
.15) A singular perturbation result for a class of periodic-parabolic BVPs  17 Finally, taking into account that the unique positive solution [ ] θ m a d , might change of sign.The main existence result for (1.1) can be stated as follows.It is Theorem 6.1 of Aleja et al. [1].Theorem 5.1.problem (1.1) admits a positive solution if, and only if, ) , we can assume, without loss of generality, that condition (3.10) is satisfied, i.e., , then (9.2) holds.and,owing to (9.4), Proposition 6.1 guarantees that, for every > ε 0, there exists ( ) > singular perturbation result for a class of periodic-parabolic BVPs  25Fix ≥ γ γ 0 .Then, due to (9.12), Proposition 5.1(iii) implies that Moreover, owing to (9.14), it follows from Proposition 6.1 that, for every > ε 0, there exists ( ) > stands for the unique positive solution of (9.18), whose existence and uniqueness follow from Theorem 5.2 applied in × .Also, by (9.10), it follows from Theorem 8.1 (or Step 2 of Theorem 8.1) applied to (9.18) that there exists ( and apply Step 4 of Theorem 8.1.So, the technical details will be omitted by repetitive.This ends the proof.□ Funding information: The authors have been supported by the Research Grants PID2021-123343NB-I00 of the Ministry of Science and Innovation of Spain and the Institute of Interdisciplinary Mathematics of Complutense University of Madrid. γ By construction, γ Similarly, if ∈x Ω\ , then, for sufficiently large > γ 0, we have that γ Therefore, for sufficiently large > γ 0, say ≥ γ γ 0 , we have that γ (9.14)A