Sharp profiles for diffusive logistic equation with spatial heterogeneity

Yan-Hua Xing; Jian-Wen Sun

doi:10.1515/ans-2022-0061

Article Open Access

Sharp profiles for diffusive logistic equation with spatial heterogeneity

Yan-Hua Xing and Jian-Wen Sun

Published/Copyright: April 13, 2023

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From the journal Advanced Nonlinear Studies Volume 23 Issue 1

Abstract

In this article, we study the sharp profiles of positive solutions to the diffusive logistic equation. By employing parameters and analyzing the corresponding perturbation equations, we find the effects of boundary and spatial heterogeneity on the positive solutions. The main results exhibit the sharp effects between boundary conditions and linear/nonlinear spatial heterogeneities on positive solutions.

Keywords: reaction–diffusion; logistic equation; heterogeneity; positive solution

MSC 2010: 35B40; 35K57; 92D25

1 Introduction and main results

It is well-known that the classical reaction–diffusion equation

(1.1) Δ u + λ m ( x ) u − c ( x ) u p ( x ) = 0 in Ω , B u = 0 on ∂ Ω

has been widely used in the study of diverse phenomena in the applied sciences (see, e.g. [2–5, 7–9,13,14,16]). In (1.1), Ω is a smooth and bounded domain in R N ( N ≥ 2 ), the constant p > 1 , m , c ∈ C 2 + σ ( Ω ¯ ) ( 0 < σ < 1 ), c ( x ) ≥ 0 for x ∈ Ω ¯ , the parameter λ > 0 , and the boundary operator B is given as follows:

B u = α ∂ u ∂ ν + β u ,

where ν is the unit outward normal to ∂ Ω and either α = 0 , β = 1 (the Dirichlet boundary condition) or α = 1 , β ≥ 0 (the Neumann or Robin boundary conditions). On the other hand, we know that both the boundary operator B and spatial heterogeneities m ( x ) and c ( x ) play a quite important role on the positive solution of (1.1). Sharp changes occur when the boundary or spatial heterogeneity are changed, see [12,16–20,22]. The simplest pattern appears in (1.1) when it reduces to the following Neumann problem:

(1.2) Δ u + λ u − u p = 0 in Ω , ∂ u ∂ ν = 0 on ∂ Ω ,

because the only positive solution of (1.2) is the constant λ 1 / ( p − 1 ) . However, when the boundary condition changes or spatial heterogeneity appears, there exists nonconstant positive solution to (1.1). Thus, it is natural to consider the problem between boundary and spatial heterogeneity, which one plays a dominating role. The main aim of this article is to investigate the sharp effect between boundary and spatial heterogeneity on the positive solution of (1.1).

In order to study the connections between (1.1) and (1.2), we consider the perturbation problem

(1.3) Δ u + ( λ + κ 1 ε α l ( x ) ) u − ( 1 + κ 2 ε β n ( x ) ) u p = 0 in Ω , ∂ u ∂ ν + κ 3 ε γ b ( x ) u = 0 on ∂ Ω ,

where the parameter ε > 0 , α > 0 denotes the quenching speed of linear spatial heterogeneity, β > 0 denotes the quenching speed of nonlinear spatial heterogeneity, and γ > 0 denotes the perturbation speed of Neumann boundary condition. As far as l ( x ) , n ( x ) , b ( x ) and constant κ i ( i = 1 , 2 , 3 ) are concerned, throughout the article, they are assumed to satisfy the following conditions:

l , n , b ∈ C 2 + σ ( Ω ¯ ) are all positive in Ω ¯ .
κ i ∈ { 0 , 1 } is nonnegative constant for i = 1 , 2 , 3 and κ 1 2 + κ 2 2 + κ 3 2 ≠ 0 .

As we have known, the existence of positive solutions to (1.3) is determined by the linear eigenvalue problem

(1.4) Δ u + ( λ + κ 1 ε α l ( x ) ) u = − ρ u in Ω , ∂ u ∂ ν + κ 3 ε γ b ( x ) u = 0 on ∂ Ω .

Let ρ ε be the unique principal eigenvalue of (1.4) for ε > 0 , we have

lim ε → 0 + ρ ε = − λ ,

see, e.g., the seminal works of Brown and Lin [6], Senn and Hess [21] and López-Gómez [15]. We then know that there exists a unique positive solution u ε ∈ C 2 + σ ( Ω ¯ ) to (1.3) for every λ > 0 , provided ε > 0 is sufficiently small, see [7,10,11].

Should it exist, the positive solution of (1.3) will be throughout denoted by u ε ( x ) in this article for ε ≥ 0 . We want to know how the spatial heterogeneity and boundary affect the nonconstant patterns (positive solutions) of (1.3). By employing parameters α , β and γ and analyzing the limiting behavior of the nonconstant solutions of (1.3) as ε → 0 , we will obtain the sharp connection between boundary condition and spatial heterogeneity. Note that a similar problem for nonlocal dispersal was studied in [23]. According to Assumption ( A 2 ) , we can see that the sharp profiles will reveal the following results:

The effect of linear/nonlinear spatial heterogeneities or Neumann boundary condition ( κ 1 2 + κ 2 2 + κ 3 2 = 1 ).
The effect between linear and nonlinear spatial heterogeneities ( κ 1 = κ 2 = 1 , κ 3 = 0 ).
The effect between linear/nonlinear spatial heterogeneities and boundary condition ( κ 1 / κ 2 = 1 and κ 3 = 1 ).
The mixed effects of spatial heterogeneities and boundary condition ( κ 1 = κ 2 = κ 3 = 1 ).

We are ready to state the main results.

Theorem 1.1

Let u ε ( x ) be the unique positive solution of (1.3) for fixed λ > 0 .

If κ 1 = 1 and one of the following assumptions holds,
1. κ 2 = κ 3 = 0 ;
2. κ 2 = κ 3 = 1 and min { β , γ } > α ;
3. κ 2 = 1 , κ 3 = 0 and β > α ;
4. κ 2 = 0 , κ 3 = 1 and γ > α ; then
lim ε → 0 + u ε ( x ) − λ 1 p − 1 ε α = U l ( x ) uniformly i n Ω ¯ ,
where U l ( x ) stands for the unique positive solution of
(1.5) Δ u + ( 1 − p ) λ u + λ 1 p − 1 l ( x ) = 0 in Ω , ∂ u ∂ ν = 0 on ∂ Ω .
If κ 2 = 1 and one of the following assumptions holds,
1. κ 1 = κ 3 = 0 ;
2. κ 1 = κ 3 = 1 and min { α , γ } > β ;
3. κ 1 = 1 , κ 3 = 0 and α > β ;
4. κ 1 = 0 , κ 3 = 1 and γ > β ; then
lim ε → 0 + λ 1 p − 1 − u ε ( x ) ε β = U n ( x ) uniformly i n Ω ¯ ,
where U n ( x ) stands for the unique positive solution of
Δ u + ( 1 − p ) λ u + λ p p − 1 n ( x ) = 0 in Ω , ∂ u ∂ ν = 0 on ∂ Ω .
If κ 3 = 1 and one of the following assumptions holds,
1. κ 1 = κ 2 = 0 ;
2. κ 1 = κ 2 = 1 and min { α , β } > γ ;
3. κ 1 = 1 , κ 2 = 0 and α > γ ;
4. κ 1 = 0 , κ 2 = 1 and β > γ ; then
lim ε → 0 + λ 1 p − 1 − u ε ( x ) ε γ = U b ( x ) uniformly i n Ω ¯ ,
where U b ( x ) stands for the unique positive solution of
Δ u + ( 1 − p ) λ u = 0 in Ω , ∂ u ∂ ν = λ 1 p − 1 b ( x ) on ∂ Ω .

According to claim (i) in Theorem 1.1, we obtain that the sharp profile is determined by the Neumann problem (1.5), when only the linear spatial perturbation appears. Thus, we know that linear spatial heterogeneity plays a key effect in (1.3) when there is no other perturbation or other perturbation has a quicker speed ( min { β , γ } > α ). Similarly, we have that the sharp profile is determined by nonlinear spatial heterogeneity (or boundary) if it has a smaller perturbation speed.

Theorem 1.2

Let u ε ( x ) be the unique positive solution of (1.3) for fixed λ > 0 .

If κ 1 = κ 2 = 1 and one of the following assumptions holds,
1. κ 3 = 0 and α = β ;
2. κ 3 = 1 and γ > β = α ; then
lim ε → 0 + u ε ( x ) − λ 1 p − 1 ε α = U l n ( x ) uniformly in Ω ¯ ,
where U l n ( x ) stands for the unique solution of
(1.6) Δ u + ( 1 − p ) λ u + λ 1 p − 1 l ( x ) − λ p p − 1 n ( x ) = 0 in Ω , ∂ u ∂ ν = 0 on ∂ Ω .
If κ 2 = κ 3 = 1 and one of the following assumptions holds,
1. κ 1 = 0 and β = γ ;
2. κ 1 = 1 and α > β = γ ; then
lim ε → 0 + λ 1 p − 1 − u ε ( x ) ε β = U n b ( x ) uniformly i n Ω ¯ ,
where U n b ( x ) is the unique positive solution of
(1.7) Δ u + ( 1 − p ) λ u + λ p p − 1 n ( x ) = 0 in Ω , ∂ u ∂ ν = λ 1 p − 1 b ( x ) on ∂ Ω .
If κ 1 = κ 3 = 1 and one of the following assumptions holds,
1. κ 2 = 0 and α = γ ;
2. κ 2 = 1 and β > α = γ ; then
lim ε → 0 + u ε ( x ) − λ 1 p − 1 ε γ = U l b ( x ) uniformly i n Ω ¯ ,
where U l b ( x ) is the unique solution of
(1.8) Δ u + ( 1 − p ) λ u + λ 1 p − 1 l ( x ) = 0 in Ω , ∂ u ∂ ν = − λ 1 p − 1 b ( x ) on ∂ Ω .

Thus, we know that the sharp profile is determined by (1.6) if both the linear and nonlinear spatial perturbations appear with an equal speed, when the boundary perturbation disappears or has a smaller speed. Similarly, we obtain that the sharp profile is determined by (1.7) (or (1.8)) if linear (nonlinear) spatial heterogeneity has a larger perturbation when the others have an equal speed. However, it is quite different to Theorem 1.1, and it follows from (1.6) and (1.8) that the indefinite sharp patterns appear since the solutions may be indefinite.

Theorem 1.3

Let u ε ( x ) be the unique positive solution of (1.3) for fixed λ > 0 . If

(1.9) κ 1 = κ 2 = κ 3 and α = β = γ ,

then

lim ε → 0 + λ 1 p − 1 − u ε ( x ) ε α = U l n b ( x ) uniformly i n Ω ¯ ,

where U l n b ( x ) stands for the unique solution of

(1.10) Δ u + ( 1 − p ) λ u − l ( x ) λ 1 p − 1 + n ( x ) λ p p − 1 = 0 in Ω , ∂ u ∂ ν = b ( x ) λ 1 p − 1 on ∂ Ω .

If (1.9) holds, we obtain the sharp profile is determined by (1.10). However, the nontrivial indefinite solution may appear in this case.

The rest of the article is organized as follows: in Section 2, we investigate the sharp profiles of (1.3) when κ 1 2 + κ 2 2 + κ 3 2 = 1 . Then, we devote Section 3 to the case κ 1 2 + κ 2 2 + κ 3 2 = 2 . Finally, we present the proofs of Theorem 1.3 in Section 4.

2 Sharp effect for κ 1 2 + κ 2 2 + κ 3 2 = 1

In this section, we consider the sharp limiting behavior of positive solutions to (1.3) when κ 1 2 + κ 2 2 + κ 3 2 = 1 . Then, we obtain the effects of boundary and linear and nonlinear spatial heterogeneities on the diffusive logistic equation.

2.1 Sharp effect of boundary

In this subsection, we consider the limiting behavior of positive solution to the following perturbation reaction–diffusion problem:

(2.1) Δ u + λ u − u p = 0 in Ω , ∂ u ∂ ν + ε γ b ( x ) u = 0 on ∂ Ω ,

where the parameter ε > 0 and the constant γ > 0 denotes the perturbation speed of boundary.

Since u ε ( x ) is nonconstant when ε > 0 , we first show that u ε ( x ) is monotone with respect to ε .

Lemma 2.1

Suppose that u ε ( x ) is the positive solution of (2.1), then u ε ( x ) is monotonically decreasing with respect to ε ≥ 0 .

Proof

For any given ε 1 > ε 2 ≥ 0 . Let u ε 1 ( x ) and u ε 2 ( x ) be the positive solutions of

(2.2) Δ u + λ u − u p = 0 in Ω , ∂ u ∂ ν + ε 1 γ b ( x ) u = 0 in ∂ Ω ,

and

Δ u + λ u − u p = 0 in Ω , ∂ u ∂ ν + ε 2 γ b ( x ) u = 0 in ∂ Ω ,

respectively. Since ε 1 > ε 2 and ε 1 γ > ε 2 γ , we have

Δ u ε 2 + λ u ε 2 − u ε 2 p = 0 in Ω , ∂ u ε 2 ∂ ν + ε 1 γ b ( x ) u ε 2 ≥ 0 on ∂ Ω .

Thus, u ε 2 ( x ) is an upper-solution of (2.2). But (2.2) admits a unique positive solution, it becomes apparent that

0 < u ε 1 ( x ) ≤ u ε 2 ( x )

for x ∈ Ω ¯ .

Now set v ( x ) = u ε 2 ( x ) − u ε 1 ( x ) , we can see that v ( x ) ≢ 0 and

Δ v − c ( x ) v ( x ) = − λ v ( x ) ≤ 0 ,

where

c ( x ) = u ε 2 p ( x ) − u ε 1 p ( x ) u ε 2 ( x ) − u ε 1 ( x ) if v ( x ) > 0 , 0 if v ( x ) = 0 .

Then, the maximum principle provides that

v ( x ) > 0 and 0 < u ε 1 ( x ) < u ε 2 ( x )

for x ∈ Ω . The proof is complete.□

Theorem 2.2

Let u ε ( x ) be the unique positive solution of (2.1), then

(2.3) lim ε → 0 u ε ( x ) = λ 1 p − 1 uniformly i n Ω ¯ .

Proof

Using Lemma 2.1, we know that

(2.4) 0 ≤ u ε ( x ) ≤ u 0 ( x ) = λ 1 p − 1

and u ε ( x ) is monotone with respect to ε . Therefore, for x ∈ Ω ¯ , the point-wise limit

U ( x ) ≔ lim ε → 0 + u ε ( x )

is well defined. In fact, we also have U ( x ) ≢ 0 and

0 ≤ U ( x ) ≤ λ 1 p − 1

for x ∈ Ω ¯ . Furthermore, we know

0 ≤ ε γ b ( x ) ≤ ε γ max Ω ¯ b ( x )

for x ∈ ∂ Ω .

On the other hand, let v ε ( x ) be the unique positive solution of

Δ u + λ u − u p = 0 in Ω , ∂ u ∂ ν + ε γ [ max Ω ¯ b ( x ) ] u = 0 on ∂ Ω ,

provided ε > 0 is small. It becomes apparent that

(2.5) 0 ≤ v ε ( x ) ≤ u ε ( x ) ≤ λ 1 p − 1

for x ∈ Ω ¯ and the point-wise limit

V ( x ) ≔ lim ε → 0 + v ε ( x )

is well defined. However, by a compactness argument involving the Schauder estimates (see e.g. [10,13,16]), we know that V ( x ) is a positive weak solution, and then a strong solution of

Δ u + λ u − u p ( x ) = 0 in Ω , ∂ u ∂ ν = 0 on ∂ Ω .

Thus, by the uniqueness of the positive solution, we must have

V ( x ) = λ 1 p − 1

for x ∈ Ω ¯ . Thanks to (2.5), we know from Dini’s theorem that (2.3) holds.□

According to (2.3), we know that the unique nonconstant positive solution converges to the positive solution with Neumann boundary condition. In order to find the sharp effect of boundary, it is interesting to establish the sharp profiles. Setting

ω ε ( x ) = λ 1 p − 1 − u ε ( x ) ε γ

for ε > 0 , we can prove the following result.

Theorem 2.3

Let U b ( x ) be the unique positive solution of

(2.6) Δ u + ( 1 − p ) λ u = 0 in Ω , ∂ u ∂ ν = λ 1 p − 1 b ( x ) on ∂ Ω .

Then, we have

lim ε → 0 + ω ε ( x ) = U b ( x ) uniformly i n Ω ¯ .

Proof

A direct computation provides that ω ε ( x ) satisfies

(2.7) Δ ω ε + λ ω ε = p θ ε p − 1 ( x ) ω ε in Ω , ∂ ω ε ∂ ν = b ( x ) u ε on ∂ Ω

for some θ ε ( x ) between u ε ( x ) and λ 1 / ( p − 1 ) . Moreover, we know from (2.4) that p θ ε p − 1 ( x ) and u ε ( x ) are bounded in L ∞ ( Ω ) , which are independent to ε . By (2.3),

lim ε → 0 + [ p θ ε p − 1 ( x ) − λ ] = ( p − 1 ) λ > 0 .

It follows from the classical elliptic estimates [1] and Sobolev imbedding theorem that there exists C > 0 , independent of ε , such that

‖ ω ε ‖ L ∞ ( Ω ) ≤ C .

Thus, by standard elliptic estimates, subject to a subsequence, still denoted by ε , there exists nonnegative U 0 ∈ L 2 ( Ω ) such that

lim ε → 0 + ω ε ( x ) = U 0 ( x ) weakly in W 1 , 2 ( Ω ) and strongly in L 2 ( Ω ) .

Using (2.7) again and following a similar argument as in the proof of Theorem 2.2, we know that U 0 ( x ) is a positive weak solution of (2.6). By elliptic regularity and maximum principle, it must be a positive strong solution. Since there exists a unique positive solution to (2.6), we obtain U 0 ( x ) = U b ( x ) for x ∈ Ω ¯ . As this argument is independent of the sequence ε , it is apparent from Sobolev imbedding theorem that

□ lim ε → 0 + ω ε ( x ) = U b ( x ) uniformly in Ω ¯ .

2.2 Sharp effect of spatial heterogeneity

In this subsection, we first study the reaction–diffusion equation

(2.8) Δ u + ( λ + ε α l ( x ) ) u − u p = 0 in Ω , ∂ u ∂ ν = 0 on ∂ Ω ,

where the parameter α > 0 denotes the perturbation speed of linear spatial heterogeneity. We can see that the effect of spatial heterogeneity disappears when ε = 0 . In this case, our main aim is to reveal the sharp effect of linear spatial heterogeneity.

Similarly to Lemma 2.1 and Theorem 2.2, we have the following result.

Lemma 2.4

Let u ε ( x ) be the positive solution of (2.8), then u ε ( x ) is monotonically increasing with respect to ε , i.e.,

u ε 2 ( x ) > u ε 1 ( x ) > 0

for x ∈ Ω ¯ and ε 2 > ε 1 ≥ 0 . Furthermore, we have

lim ε → 0 u ε ( x ) = λ 1 p − 1 uniformly i n Ω ¯ .

Now let us consider the sharp profile of positive solution u ε ( x ) to (2.8). Setting

ω ε ( x ) = u ε ( x ) − λ 1 p − 1 ε α ,

we have the following result.

Theorem 2.5

Let U l ( x ) be the unique positive solution of

(2.9) Δ u + ( 1 − p ) λ u + λ 1 p − 1 l ( x ) = 0 in Ω , ∂ u ∂ ν = 0 on ∂ Ω .

Then, we have

(2.10) lim ε → 0 + ω ε ( x ) = U l ( x ) uniformly i n Ω ¯ .

Proof

We can see that ω ε ( x ) satisfies

Δ ω ε + λ ω ε − p θ ε p − 1 ( x ) ω ε + l ( x ) u ε ( x ) = 0 in Ω , ∂ ω ε ∂ ν = 0 on ∂ Ω

for some θ ε ( x ) between λ 1 / ( p − 1 ) and u ε ( x ) . However, we know that p θ ε p − 1 ( x ) and u ε ( x ) are bounded in L ∞ ( Ω ) , which are independent to ε . By standard elliptic estimates and Sobolev imbedding theorem, we know that (2.10) holds, since there exists a unique positive solution to (2.9) [1].□

At last, we consider the reaction–diffusion equation

(2.11) Δ u + λ u − ( 1 + ε β n ( x ) ) u p = 0 in Ω , ∂ u ∂ ν = 0 on ∂ Ω ,

where the parameter ε > 0 and the constant β > 0 denotes the perturbation speed of nonlinear spatial heterogeneity.

Then we have the following result; the proof is similar to Lemma 2.1.

Lemma 2.6

Let u ε ( x ) be the positive solution of (2.11), then u ε ( x ) is monotonically decreasing with respect to ε , i.e.,

u ε 2 ( x ) > u ε 1 ( x ) > 0

for x ∈ Ω ¯ and ε 1 > ε 2 ≥ 0 . Furthermore, we have

lim ε → 0 u ε ( x ) = λ 1 p − 1 uniformly i n Ω ¯ .

Finally, it is apparent that we have the result that is similar to Theorem 2.5.

Theorem 2.7

Let U n ( x ) be the unique positive solution of

(2.12) Δ u + ( 1 − p ) λ u + λ p p − 1 n ( x ) = 0 in Ω ∂ u ∂ ν = 0 on ∂ Ω ,

we have

lim ε → 0 + λ 1 p − 1 − u ε ( x ) ε β = U n ( x ) uniformly i n Ω ¯ .

3 Sharp effect for κ 1 2 + κ 2 2 + κ 3 2 = 2

In this section, we consider the sharp limiting behavior of positive solutions to (1.3) when κ 1 2 + κ 2 2 + κ 3 2 = 2 . Then, we obtain the mixed effects between the boundary and linear/nonlinear spatial heterogeneities as well as mixed effects between linear and nonlinear spatial heterogeneities on the diffusive logistic equation.

3.1 Linear spatial heterogeneity vs boundary

By the result of previous section, we obtain the sharp changes of positive solutions of reaction–diffusion equations when the spatial heterogeneity or the boundary degenerates to the initial problem with Neumann problem. In this subsection, we shall further establish the sharp effect between spatial heterogeneity and boundary. To do this, we consider the following problem:

(3.1) Δ u + ( λ + ε α l ( x ) ) u − u p = 0 in Ω , ∂ u ∂ ν + ε γ b ( x ) u = 0 on ∂ Ω ,

where the parameter ε > 0 and α and γ are positive constants.

It is apparent that the positive solution of (3.1) may not monotone with respect to ε , since l ( x ) and b ( x ) are positive. We prove the convergence of solution by comparison method.

Theorem 3.1

Let u ε ( x ) be the unique positive solution of (3.1), then

(3.2) lim ε → 0 u ε ( x ) = λ 1 p − 1 uniformly i n Ω ¯ .

Proof

Let U ε ( x ) and V ε ( x ) be the unique solution of (2.1) and (2.8) for ε > 0 sufficiently small, respectively. Observe that

Δ U ε + [ λ + ε α l ( x ) ] U ε − U ε p ≥ 0 in Ω , ∂ U ε ∂ ν + ε γ b ( x ) U ε = 0 on ∂ Ω ,

and

Δ V ε + [ λ + ε α l ( x ) ] V ε − V ε p = 0 in Ω , ∂ V ε ∂ ν + ε γ b ( x ) V ε ≥ 0 on ∂ Ω .

Since u ε ( x ) is the unique positive solution of (3.1), we must have

U ε ( x ) ≤ u ε ( x ) ≤ V ε ( x )

for x ∈ Ω ¯ . Therefore, we know from Theorem 2.2 and Lemma 2.4 that (3.2) holds.□

Letting η = min { α , γ } and

ω ε ( x ) = u ε ( x ) − λ 1 p − 1 ε η .

We are ready to obtain the main result of this subsection.

Theorem 3.2

Suppose that α > 0 and γ > 0 , we have the following results.

If α > γ , then
lim ε → 0 + ω ε ( x ) = − U b ( x ) uniformly i n Ω ¯ ,
where U b ( x ) stands for the unique positive solution of (2.6).
If γ > α , then
lim ε → 0 + ω ε ( x ) = U l ( x ) uniformly i n Ω ¯ ,
where U l ( x ) stands for the unique positive solution of (2.9).
If α = γ , then
lim ε → 0 + ω ε ( x ) = U l b ( x ) uniformly i n Ω ¯ ,
where U l b ( x ) is the unique solution of
(3.3) Δ u + ( 1 − p ) λ u + λ 1 p − 1 l ( x ) = 0 in Ω , ∂ u ∂ ν = − λ 1 p − 1 b ( x ) on ∂ Ω .

Proof

It becomes apparent that ω ε ( x ) satisfies

Δ ω ε + λ ω ε − p θ ε p − 1 ( x ) ω ε + ε α − η l ( x ) u ε ( x ) = 0 in Ω , ∂ ω ε ∂ ν + ε γ − η b ( x ) u ε = 0 on ∂ Ω

for some θ ε ( x ) between λ 1 / ( p − 1 ) and u ε ( x ) . Since p θ ε p − 1 ( x ) and u ε ( x ) are bounded in L ∞ ( Ω ) , which are independent to ε , a similar argument as in the proof of Theorem 2.2 gives that (i)–(iii) hold.□

The interesting results (i) and (ii) of Theorem 3.2 reveal that the limiting behavior of the positive solution of (3.1) is determined by the smaller speed of spatial heterogeneity and boundary. However, the sharp change occurs when the speeds of spatial heterogeneity and boundary are equal.

3.2 Nonlinear spatial heterogeneity vs boundary

We consider the reaction–diffusion equation

(3.4) Δ u + λ u − ( 1 + ε β n ( x ) ) u p = 0 in Ω , ∂ u ∂ ν + ε γ b ( x ) u = 0 on ∂ Ω .

First, we have the following result; the proof is similar to Lemma 2.4.

Lemma 3.3

The unique positive solution u ε ( x ) of (3.4) is monotonically decreasing with respect to ε , i.e.,

u ε 2 ( x ) > u ε 1 ( x ) > 0

for x ∈ Ω ¯ and ε 1 > ε 2 ≥ 0 . Furthermore, we have

lim ε → 0 u ε ( x ) = λ 1 p − 1 uniformly i n Ω ¯ .

Letting η = min { β , γ } and

ω ε ( x ) = λ 1 p − 1 − u ε ( x ) ε η .

We are ready to give the main result of this subsection.

Theorem 3.4

Suppose that β > 0 and γ > 0 , we have the following results.

If β > γ , then
lim ε → 0 + ω ε ( x ) = U b ( x ) uniformly i n Ω ¯ ,
where U b ( x ) stands for the unique positive solution of (2.6).
If γ > β , then
lim ε → 0 + ω ε ( x ) = U n ( x ) uniformly i n Ω ¯ ,
where U n ( x ) stands for the unique positive solution of (2.12).
If β = γ , then
lim ε → 0 + ω ε ( x ) = U n b ( x ) uniformly i n Ω ¯ ,
where U n b ( x ) is the unique positive solution of
(3.5) Δ u + ( 1 − p ) λ u + λ p p − 1 n ( x ) = 0 in Ω ∂ u ∂ ν = λ 1 p − 1 b ( x ) on ∂ Ω .

Proof

A direct computation gives that ω ε ( x ) satisfies

Δ ω ε + λ ω ε − p θ ε p − 1 ( x ) ω ε + ε β − η n ( x ) u ε ( x ) = 0 in Ω , ∂ ω ε ∂ ν − ε γ − η b ( x ) u ε = 0 on ∂ Ω

for some θ ε ( x ) between λ 1 / ( p − 1 ) and u ε ( x ) . Using a similar argument as in the proof of Theorem 2.2, we know that (i)–(iii) hold.□

3.3 Linear vs nonlinear spatial heterogeneities

In this subsection, we study the sharp behavior of reaction–diffusion equation

(3.6) Δ u + ( λ + ε α l ( x ) ) u − ( 1 + ε β n ( x ) ) u p = 0 in Ω , ∂ u ∂ ν = 0 on ∂ Ω ,

where α and β are positive constants.

Since the nonconstant positive solution of (3.6) may not monotone with respect to ε , we first obtain the convergence of solution by comparison method.

Theorem 3.5

Let u ε ( x ) be the unique positive solution of (3.6), then

(3.7) lim ε → 0 u ε ( x ) = λ 1 p − 1 uniformly i n Ω ¯ .

Proof

Let U ε ( x ) and V ε ( x ) be the unique solutions of (2.8) and (2.11) for ε > 0 sufficiently small, respectively. Then, we know that U ε ( x ) is an upper-solution and V ε ( x ) is a lower-solution to (3.6). Hence, we have

V ε ( x ) ≤ u ε ( x ) ≤ U ε ( x )

for x ∈ Ω ¯ . It follows from Lemmas 2.4 and 2.6 that (3.7) holds.□

Letting η = min { α , β } and

ω ε ( x ) = u ε ( x ) − λ 1 p − 1 ε η .

We are ready to obtain the main result of this subsection.

Theorem 3.6

Suppose that α > 0 and β > 0 , we have the following results.

If α > β , then
lim ε → 0 + ω ε ( x ) = − U n ( x ) uniformly i n Ω ¯ ,
where U n ( x ) stands for the unique positive solution of (2.12).
If β > α , then
lim ε → 0 + ω ε ( x ) = U l ( x ) uniformly i n Ω ¯ ,
where U l ( x ) stands for the unique positive solution of (2.9).
If α = β , then
lim ε → 0 + ω ε ( x ) = U l n ( x ) uniformly i n Ω ¯ ,
where U l n ( x ) is the unique solution of
(3.8) Δ u + ( 1 − p ) λ u + λ 1 p − 1 l ( x ) − λ p p − 1 n ( x ) = 0 in Ω , ∂ u ∂ ν = 0 on ∂ Ω .

Proof

We can see that

Δ ω ε + λ ω ε − p θ ε p − 1 ( x ) ω ε + ε α − η l ( x ) u ε ( x ) − ε β − η n ( x ) u ε p ( x ) = 0 in Ω , ∂ ω ε ∂ ν = 0 on ∂ Ω

for some θ ε ( x ) between λ 1 / ( p − 1 ) and u ε ( x ) . Following a standard argument as in the proof of Theorem 2.2, we know that (i)–(iii) hold.□

4 Sharp effect for κ 1 2 + κ 2 2 + κ 3 2 = 3

This section is concerned with the sharp profile of positive solution to

(4.1) Δ u + ( λ + ε α l ( x ) ) u − ( 1 + ε β n ( x ) ) u p = 0 in Ω , ∂ u ∂ ν + ε γ b ( x ) u = 0 on ∂ Ω ,

where α , β and γ are positive constants.

Theorem 4.1

Let u ε ( x ) be the unique positive solution for fixed λ > 0 provided ε > 0 is small, then

(4.2) lim ε → 0 u ε ( x ) = λ 1 p − 1 uniformly i n Ω ¯ .

Proof

Let V ε ( x ) and U ε ( x ) be the unique positive solutions of (3.4) and (3.6) for ε > 0 sufficiently small, respectively. Then, we know that U ε ( x ) is an upper-solution and V ε ( x ) is a lower-solution to (4.1). Since

lim ε → 0 U ε ( x ) = lim ε → 0 V ε ( x ) = λ 1 p − 1 uniformly in Ω ¯ ,

we know that (4.2) holds.□

By a standard compact method as in the proof of Theorem 2.3, we then can show the following results, we omitting the details.

Theorem 4.2

Let u ε ( x ) be the unique positive solution of (4.1) for fixed λ > 0 .

If min { α , β } > γ , then
lim ε → 0 + λ 1 p − 1 − u ε ( x ) ε γ = U b ( x ) uniformly i n Ω ¯ ,
where U b ( x ) stands for the unique positive solution of (2.6).
If min { β , γ } > α , then
lim ε → 0 + u ε ( x ) − λ 1 p − 1 ε α = U l ( x ) uniformly i n Ω ¯ ,
where U l ( x ) stands for the unique positive solution of (2.9).
If min { α , γ } > β , then
lim ε → 0 + λ 1 p − 1 − u ε ( x ) ε β = U n ( x ) uniformly i n Ω ¯ ,
where U n ( x ) stands for the unique positive solution of (2.12).

Theorem 4.3

Let u ε ( x ) be the unique positive solution of (4.1) for fixed λ > 0 .

If α > β = γ , then
lim ε → 0 + λ 1 p − 1 − u ε ( x ) ε γ = U n b ( x ) uniformly i n Ω ¯ ,
where U n b ( x ) stands for the unique positive solution of (3.5).
If β > α = γ , then
lim ε → 0 + u ε ( x ) − λ 1 p − 1 ε γ = U l b ( x ) uniformly i n Ω ¯ ,
where U l b ( x ) stands for the unique positive solution of (3.3).
If γ > α = β , then
lim ε → 0 + u ε ( x ) − λ 1 p − 1 ε α = U l n ( x ) uniformly i n Ω ¯ ,
where U l n ( x ) stands for the unique positive solution of (3.8).

Theorem 4.4

Let u ε ( x ) be the unique positive solution of (4.1) for fixed λ > 0 . If α = β = γ , then

lim ε → 0 + λ 1 p − 1 − u ε ( x ) ε γ = U l n b ( x ) uniformly i n Ω ¯ ,

where U l n b ( x ) stands for the unique positive solution of

Δ u + ( 1 − p ) λ u − l ( x ) λ 1 p − 1 + n ( x ) λ p p − 1 = 0 in Ω , ∂ u ∂ ν = b ( x ) λ 1 p − 1 on ∂ Ω .

At the end of this section, we conclude the proofs of Theorems 1.1–1.3.

Theorem 1.1: The first claim is followed by Theorem 2.5, (ii) of Theorem 3.2, (ii) of Theorem 3.6, and (ii) of Theorem 4.2. The second claim is followed by Theorem 2.7, (ii) of Theorem 3.4, (i) of Theorem 3.6, and (iii) of Theorem 4.2. The third claim is followed by Theorem 2.3, (i) of Theorem 3.2, (i) of Theorem 3.4, and (i) of Theorem 4.2.

Theorem 1.2: The first claim is followed by (iii) of Theorem 3.6 and (iii) of Theorem 4.3. The second claim is followed by (iii) of Theorem 3.4 and (i) of Theorem 4.3. The third claim is followed (iii) of Theorem 3.2 and (ii) of Theorem 4.3.

Finally, we know from Theorem 4.4 that the conclusion of Theorem 1.3 is true.

Acknowledgments

The author would like to thank the anonymous referee for the careful reading and several valuable comments to revise the article.

Funding information: This work was partially supported National Science Foundation (No. 12271226).
Conflict of interest: There is no conflicts of interest to this work.

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Received: 2022-06-20

Revised: 2023-03-01

Accepted: 2023-03-02

Published Online: 2023-04-13

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/ans-2022-0061

Keywords for this article

reaction–diffusion; logistic equation; heterogeneity; positive solution

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