Protection Zones in Periodic-Parabolic Problems

1 Introduction

The main goal of this paper is analyzing the behavior of the principal eigenvalue Σ⁢(λ) and the associated principal eigenfunction φλ as λ↑∞, of the (linear) periodic-parabolic spectral problem

under the following general assumptions:

1. Ω is a bounded subdomain (open and connected set) of ℝN, N≥1, of class 𝒞2+θ for some 0<θ≤1, whose boundary ∂⁡Ω consists of two disjoint open and closed subsets Γ0 and Γ1 such that ∂⁡Ω:=Γ0∪Γ1 (as they are disjoint, Γ0 and Γ1 must be of class 𝒞2+θ); ∂⁡Ω cannot be connected if both Γi≠∅, i=0,1.

2. For a given T>0, 𝔏 is a non-autonomous differential operator of the form

   (1.2) 𝔏 := 𝔏 ⁢ ( x , t ) := - ∑ i , j = 1 N a i ⁢ j ⁢ ( x , t ) ⁢ ∂ 2 ∂ ⁡ x i ⁢ ∂ ⁡ x j + ∑ j = 1 N b j ⁢ ( x , t ) ⁢ ∂ ∂ ⁡ x j + c ⁢ ( x , t )

   with ai⁢j=aj⁢i,bj,c∈F for all 1≤i,j≤N, where

   F := { u ∈ 𝒞 θ , θ 2 ⁢ ( Ω ¯ × ℝ ; ℝ ) : u ⁢ ( ⋅ , T + t ) = u ⁢ ( ⋅ , t ) ⁢  for all  ⁢ t ∈ ℝ } .

   Moreover, we assume that 𝔏 is uniformly elliptic in Q¯T=Ω¯×[0,T], i.e., there exists μ>0 such that

   ∑ i , j = 1 N a i ⁢ j ⁢ ( x , t ) ⁢ ξ i ⁢ ξ j ≥ μ ⁢ | ξ | 2   for all  ⁢ ( x , t , ξ ) ∈ Q ¯ T × ℝ N ,

   where |⋅| stands for the Euclidean norm of ℝN.

3. 𝔅 ≡ 𝔅 ⁢ [ β ] : 𝒞 ⁢ ( Γ 0 ) ⊕ 𝒞 1 ⁢ ( Ω ∪ Γ 1 ) → C ⁢ ( ∂ ⁡ Ω ) stands for the boundary operator

   𝔅 ⁢ ξ := { ξ on  ⁢ Γ 0 , ∂ ⁡ ξ ∂ ⁡ ν + β ⁢ ( x ) ⁢ ξ on  ⁢ Γ 1 ,   ξ ∈ 𝒞 ⁢ ( Γ 0 ) ⊕ 𝒞 1 ⁢ ( Ω ∪ Γ 1 ) ,

   where β∈𝒞1+θ⁢(Γ1) and ν=(ν1,…,νN)∈𝒞1+θ⁢(∂⁡Ω;ℝN) is an outward pointing nowhere tangent vector field. When Γ1=∅, we will set 𝔇=𝔅 because 𝔅 becomes the Dirichlet boundary operator.

4. m ∈ F satisfies m⪈0 and, eventually,

   (1.3) m ⁢ ( x , t ) > 0   for all  ⁢ ( x , t ) ∈ ∂ ⁡ Ω × [ 0 , T ] .

Under these general conditions, it is folklore that any solution φ of (1.1) lies in the Banach space of Hölder continuous T-periodic functions

Throughout this paper, we will also consider the associated periodic-parabolic operator

The existence of a principal eigenvalue for (1.1) in the special case when β≥0 goes back to Beltramo and Hess [7], while it is a recent result of [5] when β changes its sign. Throughout this paper, the principal eigenvalue of (1.1) will be denoted by

to emphasize its dependence on the real parameter λ∈ℝ. It is unique and algebraically simple [5]. By the monotonicity of the principal eigenvalue with respect to the potential (see Lemma 2.2), it becomes apparent that Σ⁢(λ) is a continuous strictly increasing function of λ because m⪈0 in QT. Therefore, the limit

is well defined. Actually, by [4, Theorem 5.1], Σ⁢(λ) is analytic and concave in λ and, owing to [4, Proposition 5.2], Σ′⁢(λ)>0 for all λ∈ℝ, where ′ stands for differentiation with respect to λ. The main goal of this paper is ascertaining whether or not Σ∞ is finite. Precisely, for a large variety of weight functions m⁢(x,t) our main result establishes the following:

1. Σ ∞ < ∞ if and only if there exists a continuous map κ:[0,T]→Ω such that (κ⁢(t),t)∈int⁡m-1⁢(0) for all t∈[0,T]. In other words, if we can advance upwards in time from t=0 up to t=T within the interior of m-1⁢(0).

Our proof of the fact that Σ∞<∞ if κ exists is based on a technical device of Beltramo and Hess [7], which can be easily adapted to sharpen, very substantially, some of the previous findings of Daners and Thornett [14]. The fact that the existence of κ is not only sufficient but also necessary is a new finding that relies on a tricky use of a novel generalized version of the parabolic maximum principle on general domains, not necessarily cylindric, which is delivered in Section 3.

These spectral problems were introduced by the author in [24], in an elliptic context, in order to characterize the existence of principal eigenvalues in weighted boundary value problems. The main result of [24] was later used by the author in [25] to show that the principle of competitive exclusion is false in the presence of protection zones for each of the species. These results and their multiple variants, collected in [26], were a milestone for the generation of new results in Spatial Ecology. Among them count, in the periodic-parabolic context, the papers of Du and Peng [15, 16] and Daners and Thornett [14], where some partial answers to the singular perturbation problem addressed in this paper were given.

The contents of this paper are the following. Section 2 collects some useful results of Antón and López-Gómez [3, 4, 5] used throughout this paper. Section 3 delivers a generalized version of the parabolic maximum principle in general domains. In the new theorem it is imperative to fix the concepts of lateral and flat top boundaries for a general domain G, not necessarily cylindric. Section 4 establishes that Σ∞<∞ if κ exists. Section 5 establishes the uniform exponential decay of φλ to zero on compact subsets of [m>0] as λ↑∞, after normalization, when Σ∞<∞. Section 6 shows that Σ∞=∞ if κ does not exist for a large class of paradigmatic m’s. Finally, in Section 7, the main theorem of Section 4 is invoked to get a permanence result in the context of periodic-parabolic competing species models, which extends, very substantially, some of the old findings of the author in [25]. According to these results, the periodic-parabolic protection zones of the species u must consist of corridors of life linking the time levels t=0 and t=T upwards in time within the interior of m-1⁢(0), i.e., m-1⁢(0) is a protection zone of u if and only if κ exists.

The results of this paper allow to characterize the asymptotic behavior of the positive periodic solutions of the next class of periodic-parabolic semilinear problems

where μ∈ℝ, q>1 and m⪈0 is T-periodic, though this analysis will be accomplished elsewhere. Kondratiev and Véron [21] made some important progress in an autonomous prototype of (1.5) under Neumann boundary conditions (Γ0=∅ and β=0) when μ=0 and m⁢(x)=m⁢(x,t) is not a positive constant. It is imperative to extend the theory of Marcus and Véron [30] to a periodic-parabolic context in order to characterize the asymptotic profiles of the solutions of (1.5) in the range of values of μ where (1.5) cannot admit any positive periodic solution. The pioneering work of Véron and his collaborators has been pivotal to characterize the dynamics of wide classes of semilinear and quasilinear problems in the presence of spatio-temporal heterogeneities (see [29] and the list of references therein).

2 Some Preliminaries

This section collects some of the most basic properties of the principal eigenvalue σ⁢[𝒫,𝔅,QT] that are going to be used in this paper. Most of them go back to [4] and are based on the next pivotal characterization, going back in the context of this paper to [4, Theorem 1.1].

The next result establishes the monotonicity of the principal eigenvalue with respect to the potential. It is [4, Proposition 2.1].

As a direct consequence of Lemma 2.2, if {Vn}n≥1 is a sequence of potentials in F converging to some V∈F in 𝒞⁢(Q¯T), then

In particular, the function Σ⁢(λ) defined in (1.4) is continuous and strictly increasing with respect to λ. Actually, by [4, Theorem 5.1], it is analytic.

The following monotonicity property extends [8, Propositions 3.1, 3.2 and 3.5] to a periodic-parabolic context. It is [4, Proposition 2.4].

The next result is [4, Proposition 2.5].

Now, suppose Γ1≠∅. Then, for every proper subdomain Ω0 of Ω of class 𝒞2+θ with

we denote by 𝔅⁢[Ω0] the boundary operator defined, for any ξ∈𝒞⁢(Γ0)⊕𝒞1⁢(Ω∪Γ1), by

In particular, 𝔅⁢[Ω0]=𝔇 if Ω¯0⊂Ω because, in such case, ∂⁡Ω0⊂Ω. When Γ1=∅, by definition, 𝔅=𝔇 and we simply set 𝔅⁢[Ω0]=𝔇. The next result establishes the monotonicity of the principal eigenvalue with respect to Ω. It is [4, Proposition 2.6].

3 A Generalized Parabolic Maximum Principle

The most classical version of the parabolic maximum principle is the one collected in, e.g., John’s book [20, Chapter 7]. According to it, for any given bounded open subset Ω of ℝN and any T>0 the following result holds in the parabolic cylinder QT=Ω×(0,T).

Theorem 3.1 has a huge number of applications to get uniqueness in a large class of parabolic problems, the uniqueness results collected in John’s textbook [20, p. 176] being a paradigm. Although there have been some attempts to extend it to more general domains than parabolic cylinders (see, e.g., [10, Chapter 3]), the class of domains G where the classical parabolic maximum principle applies always have a very special geometry looking like a parabolic cylinder. However, in a huge number of applications it is highly desirable to have a formulation of it applicable in arbitrary open bounded subsets G of ℝN×ℝ. The main goal of this section is providing one of these generalizations.

Throughout this section, for any given open connected subset G⊂ℝN×ℝ with N≥1, the concepts introduced in the next definition will be adopted. Subsequently, we denote by (x,t) the points of ℝN×ℝ with x∈ℝN. Moreover, for any given x∈ℝN and R>0, we denote by BR⁢(x) the open ball of ℝN centered at x of radius R, i.e.,

In Definition 3.2 (b), Cε⁢(x,t) stands for the cylinder with section Bε⁢(x) and central axis {x}×(t-ε,t). Figure 1 shows an admissible non-simply connected domain G together with the components of ∂F⁢T⁡G, labeled by ℱi, 1≤i≤7. According to Definition 3.2 (a), ∂F⁢T⁡G is an open subset of ∂⁡G, and the lateral boundary ∂L⁡G is a compact subset of G¯ because G is bounded. In Figure 1, ∂L⁡G has been plotted using a thicker continuous line than for the components of ∂F⁢T⁡G. In total, ∂F⁢T⁡G and ∂L⁡G consist of seven components each. The next result is a natural extension of Theorem 3.1.

Theorem 3.3 admits a natural extension to cover a general class of second-order non-autonomous uniformly elliptic operators of the form

with

for which there exists a constant μ>0 such that

Under these assumptions, the following generalized version of Theorem 3.3 holds. It is the main result of this section.

Figure 1

An admissible open domain G, together with the components of the flat top boundary ∂F⁢T⁡G andthe lateral boundary ∂L⁡G.

Theorem 3.4 is a direct consequence of Theorem 3.3. Thus, we begin by proving Theorem 3.3.

3.1 Proof of Theorem 3.3

The following concept plays a crucial role in the proof.

Definition 3.5.

Let ℱ be an open and connected component of ∂F⁢T⁡G containing (x,t). Then:

1. ℱ is said to be of type ⊓ if there exists ε>0 such that

   [ ( ℱ + B ε ⁢ ( 0 ) ) × [ t , t + ε ] ] ∩ G = ∅ .

2. ℱ is said to be of type ⊔ if there exists ε>0 such that

   [ ( ℱ + B ε ⁢ ( 0 ) ) × ( t - ε , t ) ] ⊂ G .

3. ℱ is said to be of type ⊥ if it is neither of type ⊓ nor of type ⊔. In other words, when, for every ε>0,

   { [ ( ℱ + B ε ⁢ ( 0 ) ) × [ t , t + ε ] ] ∩ G ≠ ∅ , [ ( ℱ + B ε ⁢ ( 0 ) ) × ( t - ε , t ) ] ∩ ( ℝ N + 1 ∖ G ) ≠ ∅ .

According to this definition, ℱ1, ℱ4 and ℱ6 are components of type ⊓ of the open set G represented in Figure 1, while ℱ3, ℱ5 and ℱ7 are components of type ⊔. The unique component of type ⊥ of Figure 1 is ℱ2.

Proof of Theorem 3.3.

The proof of Theorem 3.3 will proceed into four steps. First, we will prove it when, instead of (3.1), the strongest condition holds:

(3.4) ∂ t ⁡ u - κ ⁢ Δ ⁢ u < 0   in  ⁢ G .

The proof of the general case will be reduced to this special case.

Step 1. Suppose (3.4) and that all the components of ∂F⁢T⁡G are of type ⊓. Figure 2 shows an admissible G whose flat top boundary consists of components of type ⊓.

Figure 2

A flat top boundary consisting of components of type ⊓.

Let ℱi, 1≤i≤q, denote the components of ∂F⁢T⁡G. Then there exist q open subsets Ωi of ℝN and q values of the temporal variable ti, 1≤i≤q, such that

ℱ i = Ω i × { t i } , 1 ≤ i ≤ q .

Since the closures of these components are assumed to be disjoint and all of them are of type ⊓, it becomes apparent that, for sufficiently small ε>0, the truncated open set

G ε = G ∖ { ( x , t ) ∈ G : dist ⁡ ( ( x , t ) , ∂ F ⁢ T ⁡ G ) < ε }

also provides us with an open subset of ℝN×ℝ whose flat top boundary consists of q components,

ℱ i , ε = Ω i , ε × { t i - ε } , 1 ≤ i ≤ q ,

such that, in the appropriate sense,

lim ε ↓ 0 ⁡ Ω i , ε = Ω i   for all  ⁢ 1 ≤ i ≤ q .

Figure 3 shows an admissible Gε. Since G¯ε⊂G¯ is compact and u∈𝒞⁢(G¯ε), there exists (xm,tm)∈G¯ε such that

u ⁢ ( x m , t m ) = max G ¯ ε ⁡ u .

Figure 3

The truncated open subset Gε.

We claim that we can take

(3.5) ( x m , t m ) ∈ ∂ L ⁡ G ε = ∂ ⁡ G ε ∖ ∂ F ⁢ T ⁡ G ε .

To show (3.5), we will argue by contradiction assuming that

(3.6) ( x m , t m ) ∈ G ε ∪ ∂ F ⁢ T ⁡ G ε .

Suppose that (xm,tm)∈Gε. Then

0 = ∇ ( x , t ) ⁡ u ⁢ ( x m , t m ) = ( ∇ x ⁡ u ⁢ ( x m , t m ) , ∂ t ⁡ u ⁢ ( x m , t m ) ) .

Moreover, since (xm,tm) is a local maximum, Δ⁢u⁢(xm,tm)≤0. Thus,

∂ t ⁡ u - κ ⁢ Δ ⁢ u | ( x m , t m ) = - κ ⁢ Δ ⁢ u ⁢ ( x m , t m ) ≥ 0 ,

which contradicts (3.4). Hence, it follows from (3.6) that

( x m , t m ) ∈ ∂ F ⁢ T ⁡ G ε = ⋃ i = 1 q ℱ i , ε .

Thus, (xm,tm)∈ℱi,ε for some i∈{1,…,q}. Consequently, since it is a local maximum and ℱi,ε is open, owing to Definition 3.2 (b), it becomes apparent that

∂ t ⁡ u ⁢ ( x m , t m ) ≥ 0 , ∇ x ⁡ u ⁢ ( x m , t m ) = 0 , Δ ⁢ u ⁢ ( x m , t m ) ≤ 0 ,

Therefore,

∂ t - κ ⁢ Δ ⁢ u | ( x m , t m ) ≥ - κ ⁢ Δ ⁢ u ⁢ ( x m , t m ) ≥ 0 ,

which also contradicts (3.4). Therefore, (3.5) holds, and hence

(3.7) u ⁢ ( x m , t m ) = max G ¯ ε ⁡ u = max ∂ L ⁡ G ε ⁡ u ≤ max ∂ L ⁡ G ⁡ u

because ∂L⁡Gε⊂∂L⁡G. By continuity, (3.7) implies that

max G ¯ ⁡ u = sup ε > 0 ⁡ max G ¯ ε ⁡ u ≤ max ∂ L ⁡ G ⁡ u .

Therefore, (3.2) holds under condition (3.4).

Step 2. Suppose (3.4) and that ∂⁡Ω has a component of type ⊔, say ℱ. Then, by Definition 3.5 (ii), there exists ε>0 such that

D := [ ( ℱ + B ε ⁢ ( 0 ) ) × ( t - ε , t ) ] ⊂ G .

Since the flat top boundary of D consists of a single component of type ⊓, by Step 1, we can infer that

(3.8) max D ¯ ⁡ u = max ∂ L ⁡ D ⁡ u .

Thus, if the point where u⁢(x,t) reaches the maximum in G¯, say (xm,tm), lies in ℱ, then, owing to (3.8), u⁢(x,t) must also reach its maximum value on ∂L⁡D⊂G (see Figure 4). By Step 1, this is impossible under condition (3.4). Therefore, maxG¯⁡u cannot be taken on any component of type ⊔.

Figure 4

The open set D in Step 2.

Step 3. Suppose (3.4) and that ∂⁡Ω has a component of type ⊥, say ℱ. Then, by Definition 3.5 (iii), we have that, for every ε>0,

{ [ ( ℱ + B ε ⁢ ( 0 ) ) × [ t , t + ε ] ] ∩ G ≠ ∅ , [ ( ℱ + B ε ⁢ ( 0 ) ) × ( t - ε , t ) ] ∩ ( ℝ N + 1 ∖ G ) ≠ ∅ .

In this case, for sufficiently small ε>0, we consider the auxiliary open subset

D := [ ( ℱ + B ε ⁢ ( 0 ) ) × ( t - ε , t ) ] ∩ G ,

which is nonempty by Definition 3.2 (see Figure 5).

Figure 5

The open set D in Step 3.

The flat top boundary of D consists of a single component of type ⊓ containing ℱ. Thus, by Step 1,

(3.9) max D ¯ ⁡ u = max ∂ L ⁡ D ⁡ u .

Suppose that the point where u⁢(x,t) reaches the maximum in G¯, say (xm,tm), lies on ∂F⁢T⁡D. Then, by (3.9), maxG¯⁡u is also taken on ∂L⁡D, which consists of points of G and of points of ∂L⁡G. Due to Step 1, we already know that, under condition (3.4), maxG¯⁡u cannot be taken in G. Therefore, it must be taken on ∂L⁡G. Hence, also in this case, one can infer from (3.9) that

max G ¯ ⁡ u = max ∂ L ⁡ G ⁡ u .

Step 4. Suppose that condition (3.1) holds. Then, for any given c>0, we can consider the auxiliary function

(3.10) v c ⁢ ( x , t ) := u ⁢ ( x , t ) - c ⁢ t ≤ u ⁢ ( x , t ) , ( x , t ) ∈ G ¯ .

Obviously, according to (3.1) and (3.10), we have that

∂ t ⁡ v c - κ ⁢ Δ ⁢ v c = ∂ t ⁡ u - κ ⁢ Δ ⁢ u - c ≤ - c < 0

for all c>0. Thus, the function vc satisfies (3.4) for all c>0. Consequently, by the result of Steps 1–3 and thanks to (3.10), we obtain that

max G ¯ ⁡ u = max G ¯ ⁡ ( v c ⁢ ( x , t ) + c ⁢ t ) ≤ max G ¯ ⁡ v c + c ⁢ T = max ∂ L ⁡ G ⁡ v c + c ⁢ T ≤ max ∂ L ⁡ G ⁡ u + c ⁢ T .

Therefore, letting c↓0 yields

max G ¯ ⁡ u ≤ max ∂ L ⁡ G ⁡ u ,

which ends the proof of the theorem. ∎

3.3 The Strong Maximum Principle Revisited

In this section, we collect the version of the strong maximum principle of Nirenberg [31] delivered in [17, Proposition 13.1] and [5, Theorem 2.1]. It extends the classical minimum principle of E. Hopf [19].

Theorem 3.6.

Let G be a bounded domain of RN×R and let L⁢(x,t) be a uniformly elliptic operator in G of the form

𝔏 := 𝔏 ⁢ ( x , t ) := - ∑ i , j = 1 N a i ⁢ j ⁢ ( x , t ) ⁢ ∂ 2 ∂ ⁡ x i ⁢ ∂ ⁡ x j + ∑ j = 1 N b j ⁢ ( x , t ) ⁢ ∂ ∂ ⁡ x j + c ⁢ ( x , t ) ,

with

a i ⁢ j = a j ⁢ i , b j , c ∈ 𝒞 θ , θ 2 ⁢ ( G ¯ )   𝑎𝑛𝑑   c ≥ 0 .

Suppose that a function u∈C2,1⁢(G)∩C⁢(G¯) satisfies

∂ t ⁡ u ⁢ ( x , t ) + 𝔏 ⁢ ( x , t ) ⁢ u ⁢ ( x , t ) ≤ 0   for all  ⁢ ( x , t ) ∈ G ,

i.e., u is sub-harmonic for the parabolic operator P:=∂t+L⁢(x,t), and

M := max G ¯ ⁡ u ≥ 0 .

Assume that the maximum M is attained at an (interior) point (x0,t0)∈G. Then the following assertions hold:

1. u = M in the (connected) component of

   G ⁢ ( t 0 ) := { ( x , t ) ∈ G : t = t 0 }

   containing ( x 0 , t 0 ) .

2. u ⁢ ( x , t ) = M if ( x , t ) ∈ G can be connected with ( x 0 , t 0 ) by a path in G consisting only of horizontal and upward vertical segments, like illustrated by Figure 6.

Figure 6

Construction of the portion of G where u≡m.

According to the proof of our Proposition 4.2 in the next section, Theorem 3.6 can be re-written as follows.

Theorem 3.7.

Under the assumptions of Theorem 3.6, condition (b) is equivalent to the following one:

1. u ⁢ ( x , t ) = M if t < t 0 and there exists a continuous function κ : [ t , t 0 ] → G such that κ ⁢ ( t ) = x and κ ⁢ ( t 0 ) = x 0 , i.e., if the points ( x 0 , t 0 ) and ( x , t ) can be connected within G by a continuous curve κ parameterized by t.

The monograph of Hess [17] and the paper of Antón and the author [5] include some well-known applications of Theorem 3.6.

4 A General Sufficient Condition for Σ∞<∞

The main result of this section reads as follows. As the condition (1.3) is unnecessary for the validity of this result, m might vanish on ∂⁡Ω×[0,T].