blow-up profiles for positive solutions of nonlocal dispersal equation

Accepted Manuscript

Blow-up profiles for positive solutions of nonlocal dispersal equation

Jian-Wen Sun, Wan-Tong Li, Fei-Ying Yang

PII: S0893-9659(14)00378-4DOI: http://dx.doi.org/10.1016/j.aml.2014.11.009Reference: AML 4668

To appear in: Applied Mathematics Letters

Received date: 21 October 2014Accepted date: 13 November 2014

Please cite this article as: J.-W. Sun, W.-T. Li, F.-Y. Yang, Blow-up profiles for positivesolutions of nonlocal dispersal equation, Appl. Math. Lett. (2014),http://dx.doi.org/10.1016/j.aml.2014.11.009

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http://dx.doi.org/10.1016/j.aml.2014.11.009

Blow-up profiles for positive solutions of nonlocal

dispersal equation

Jian-Wen Sun∗, Wan-Tong Li and Fei-Ying Yang

School of Mathematics and Statistics,

Key Laboratory of Applied Mathematics and Complex Systems,

Lanzhou University, Lanzhou, 730000, PR China

October 21, 2014

Abstract

In this paper, we study the blow-up profiles of the nonlocal dispersal equation.More precisely, we prove that the positive solution of nonlocal dispersal equation hasdifferent blow-up profiles, depending on the refuge domain.

Keywords: Nonlocal dispersal; positive solution; blow-up.AMS Subject Classification (2010): 35B40, 35K57, 92D25

∗Corresponding author ([email protected]; [email protected]).

1

*ManuscriptClick here to view linked References

2 Blow-up profiles for positive solutions of nonlocal dispersal equation

1 Introduction and main results

In this paper, we consider the nonlocal dispersal equation

J ∗ u(x)− u(x) = −λu(x) + a(x)up(x), x ∈ Ω,

u(x) = 0, x ∈ RN \ Ω,(1.1)

where Ω is a smooth bounded domain of RN , p > 1 and λ is a real parameter. Thefunction J is continuous and

D[u](x) = J ∗ u(x)− u(x) =∫

RN

J(x− y)u(y)dy − u(x)

denotes a nonlocal dispersal operator. The coefficient a ∈ C(Ω) is nonnegative. Through-out this paper, we make the following assumptions:

(A1) J ∈ C(RN) verifies J > 0 in B1 (the unit ball), J = 0 in RN \ B1, J(x) = J(−x)with

∫RN J(x)dx = 1.

(A2) a(x) ∈ C(Ω), a(x) 6≡ 0 and vanishes in a smooth subdomain Ω0 of Ω.

Equation (1.1) has been used to model different diffusion phenomena in the literatureand attracted considerable interests, for example, the papers [1–4,6,9,11,12]. In fact, theprincipal eigenvalue of the nonlocal equation

J ∗ u− u = −λu, x ∈ Ω,

u = 0, x ∈ RN \ Ω(1.2)

is useful in the study of positive solutions to (1.1). It follows from [8] that (1.2) admitsa unique principal eigenvalue λP (Ω) associated with a positive eigenfunction and 0 <

λP (Ω) < 1. Particularly, the positive solution of (1.1) is well studied, see the work ofGarcıa-Melian and Rossi [7]. An important result is as follows.

Theorem 1.1 There exists a positive solution uλ of (1.1) if and only if λP (Ω) < λ <

λP (Ω0). In that case, uλ ∈ C(Ω), it is unique, increasing with respect to λ and verifies

limλ→λP (Ω)+

uλ(x) = 0 uniformly in Ω

and

limλ→λP (Ω0)−

uλ(x) = ∞ uniformly in Ω.

Sun and Li 3

In order to reveal the complex influence of heterogeneous environment on the positivesolutions of (1.1), in this paper we consider the nonlocal dispersal equation

J ∗ u(x)− u(x) = −λu(x) + [a(x) + ε]up(x), x ∈ Ω,

u(x) = 0, x ∈ RN \ Ω,(1.3)

where ε > 0 is a small perturbation parameter. We know that (1.3) admits a uniquepositive solution uε for every λ > λP (Ω).

Under the hypotheses of (A1) − (A2), we can prove the following conclusions.

Theorem 1.2 Let uε ∈ C(Ω) be the positive solution of (1.3) for λ > λP (Ω).

(i) If λP (Ω) < λ < λP (Ω0), then

limε→0

uε(x) = u(x) uniformly in Ω,

where u is the unique positive solution of (1.1).

(ii) If λ > λP (Ω0), thenlimε→0

uε(x) = ∞ uniformly in Ω.

The result (ii) of Theorem 1.2 shows that uε goes to infinity as ε → 0. It is natural toask how the blow-up profiles of uε can exist. In the present paper, we will further studythe blow-up profiles of uε and establish the main results as follows.

Theorem 1.3 Assme that uε ∈ C(Ω) is the positive solution of (1.3) for λ > λP (Ω). Letvε = ε

1p−1 uε and ωε = ε

1p(p−1) uε, we have the following results.

(i) If λP (Ω) < λ < λP (Ω0), then

limε→0

vε = limε→0

ωε = 0 uniformly in Ω.

(ii) If λ > λP (Ω0), then

(ii-a) limε→0 vε(x) = θ(x) uniformly in Ω0 and

(ii-b)

limε→0

ωε(x) =

[∫Ω0

J(x− y)θ(y)dy

a(x)

] 1p

uniformly in any compact subset of Ω\Ω0,

where θ satisfies

J ∗ θ(x)− θ(x) = −λθ(x) + θp(x), x ∈ Ω0,

θ(x) > 0, x ∈ Ω0,

θ(x) = 0, x ∈ RN \ Ω0.


Remark 1.4 From Theorem 1.3, we know that the blow-up profiles of positive solutionsto (1.3) changes in Ω, depending on the refuge domain Ω0. On the other hand, (1.3) canbe considered as the nonlocal analogue of the local diffusion equation

∆v(x) = −λv(x) + [a(x) + ε]vp(x), x ∈ Ω,

v(x) = 0, x ∈ ∂Ω.(1.4)

The structure of positive solutions to (1.4) is well understood [5]. In fact, the positivesolution vε of (1.4) only blows up in Ω0 as ε → 0 and vε has the same blow-up profiles inΩ0. These results are quite different from the nonlocal dispersal problem (1.3).

The rest of this paper is organized as follows. In Section 2, we give some preliminariesand prove Theorem 1.2. In Section 3, we consider the blow-up profile of (1.3).

2 Preliminaries

Since uε ∈ C(Ω) is the positive solution of (1.3) for λ > λP (Ω) and vε = ε1

p−1 uε, wehave

J ∗ vε(x)− vε(x) = −λvε(x) + [a(x)ε + 1]vp

ε (x), x ∈ Ω,

vε(x) = 0, x ∈ RN \ Ω.(2.1)

Denote Ω = Ω1, then we know that for every λ > λP (Ωi), there exists a unique solutionθi satisfies

J ∗ θi(x)− θi(x) = −λθi(x) + θpi (x), x ∈ Ωi,

θi(x) > 0, x ∈ Ωi,

θi(x) = 0, x ∈ RN \ Ωi,

(2.2)

here i = 0, 1.

To begin with, we give some estimates of vε.

Lemma 2.1 Let θi be the solution of (2.2) for λ > λP (Ωi), i = 0, 1. Then we have

vε(x) ≤ θ1(x) in Ω (2.3)

and

vε(x) ≥ θ0(x) in Ω0 (2.4)

for ε > 0.

Sun and Li 5

Proof. Since ∫

ΩJ(x− y)vε(y)dy − vε(x) + λvε(x)− vp

ε(x)

=a(x)

εvpε(x) ≥ 0

for x ∈ Ω, we have vε is a lower-solution of (2.2) (i = 1). A standard comparison argumentwith the uniqueness of the solution of (2.2) shows that (2.3) holds.

Similarly, we have∫

Ω0

J(x− y)vε(y)dy − vε(x) + λvε(x)− vpε(x)

=−∫

Ω\Ω0

J(x− y)vε(y)dy ≤ 0

for x ∈ Ω0 and (2.4) holds. 2

Lemma 2.2 Assume that λ > λP (Ω0). Then there exists a δ > 0 such that

1− λ + vp−1ε (x) ≥ δ in Ω0

for ε > 0.

Proof. If there exist εn > 0 and x∗ ∈ Ω0 such that

1− λ + vp−1εn

(x∗) ≤1n

for all n ≥ 1, then we know from (2.1) and (2.3) that

0 ≤∫

ΩJ(x− y)vεn(y)dy ≤ θ1(x∗)

nin Ω0.

So subject to a subsequence, still denoted by εn, there exists v ∈ L1(Ω) such that

limn→∞

∫

ΩJ(x− y)vεn(y)dy =

∫

ΩJ(x− y)v(y)dy = 0,

and we have v(x) = 0 in L1(Ω). Since maxΩ0vεn(x) = vεn(xn) ≥ maxΩ0

θ0(x) = θ0(x∗)for some x∗, xn ∈ Ω0, we have

‖J‖∞∫

Ωvεn(y)dy ≥

∫

ΩJ(xn − y)vεn(y)dy ≥ [1− λ + θp−1

0 (x∗)]θ0(x∗). (2.5)

But

[1− λ + θp−10 (x∗)]θ0(x∗) =

∫

Ω0

J(x∗ − y)θ0(y)dy > 0,


then it follows from (2.5) that

[1− λ + θp−10 (x∗)]θ0(x∗) = 0,

which is a contradiction. 2The proof of Theorem 1.2 is obtained by standard super-lower solutions method and

steps discussions, see [7, 10], but we still give the main process for completeness.Proof. (i) Since the positive solution of (1.3) is unique, we can see that uε increase inthe sense that uε1 < uε2 in Ω if ε1 > ε2. With the aid of dominated convergence theorem,we have limε→0 uε(x) is a positive solution of (1.1). Thanks to the uniqueness result andDini’s Theorem, we have limε→0 uε(x) = u(x) uniformly in Ω.

(ii) We can see that∫Ω uε(x)dx → ∞ as ε → 0. Otherwise, we have that u1 =

limδ→0 vδ ∈ L1(Ω) by the monotone convergence theorem. Hence u1 is bounded almosteverywhere and is a weak solution of (1.1) with λ > λP (Ω0), which is impossible ( see [7]).Then following the arguments of [7], it is easy to show that

limε→0

uε(x) = ∞ uniformly in Ω

after finitely steps as Ω is bounded. 2

3 Proof of Theorem 1.3

In this section, we give the proof of our main theorem. The first claim of Theorem 1.3is followed by Theorem 1.2. We only need to prove claim (ii).

Since vε = ε1

p−1 uε, we have∫

ΩJ(x− y)vε(y)dy − vε(x) = −λvε(x) + vp

ε(x) in x ∈ Ω0.

From Lemmas 2.1 and 2.2, there holds

|vε(x1)− vε(x2)| ≤maxΩ θ1(x)

δ

∫

Ω|J(x1 − y)− J(x2 − y)|dy

for any x1, x2 ∈ Ω0. It follows from (2.3)-(2.4) and the compactness argument that thereexits v′ such that v′(x) > 0 in Ω0 and

limε→0

vε(x) = v′(x) uniformly in Ω0. (3.1)

On the other hand, since∫

ΩJ(x−y)vε(y)dy−vε(x) = −λvε(x)+

[a(x)

ε+ 1

]vpε(x) in any compact subset of Ω\Ω0,

Sun and Li 7

we know from (2.3) that

0 ≤ vε(x) ≤[

(λ + 2)maxΩ θ1(x)a(x)

ε + 1

] 1p

.

Thus we have

limε→0

vε(x) = 0 uniformly in any compact subset of Ω \ Ω0. (3.2)

In view of (3.1) and (3.2), by dominated convergence theorem, we have

J ∗ v′(x)− v′(x) = −λv′(x) + v′p(x), x ∈ Ω0,

v′(x) > 0, x ∈ Ω0,

v′(x) = 0, x ∈ RN \ Ω0.

(3.3)

But (3.3) admits a unique solution for λ > λP (Ω), we have v′(x) = θ(x) in Ω0, this givesthe proof of (ii-a).

Now we prove (ii-b). We can see that ωε = ε1

p(p−1) uε satisfies∫

ΩJ(x− y)ωε(y)dy − ωε(x) = −λωε(x) + [a(x) + ε]

ωpε(x)ε1/p

in Ω.

Then we get∫

ΩJ(x− y)vε(y)dy − vε(x) = −λvε(x) + [a(x) + ε]ωp

ε(x)

for x of any compact subset of Ω \ Ω0 and so

ωε(x) =[∫

Ω J(x− y)vε(y)dy − vε(x) + λvε(x)a(x) + ε

] 1p

.

Thanks to (3.1)-(3.2), we know that

limε→0

ωε(x) =

[∫Ω0

J(x− y)θ(y)dy

a(x)

] 1p

uniformly in any compact subset of Ω \ Ω0.

Acknowledgments

The authors would like to thank the anonymous reviewer for his/her helpful comments.The first author was supported by NSF of China (11401277) and FRFCU (lzujbky-2014-23) and the second author was supported by NSF of China (11271172).


References

[1] P. Bates, G. Zhao, Existence, uniquenss, and stability of the stationary solution toa nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl.332 (2007) 428–440.

[2] A. Capella, J. Davila, L. Dupaigne, Y. Sire, Regularity of radial extremal solutionsfor some non-local semilinear equations, Comm. Partial Differential Equations 36(2011) 1353–1384.

[3] J. Coville, On a simple criterion for the existence of a principal eigenfunction of somenonlocal operators, J. Differential Equations 249 (2010) 2921–2953.

[4] J. Coville, J. Davila, S. Martınez, Existence and uniqueness of solutions to nonlocalequation with monostable nonlinearity, SIAM J. Math. Anal. 39 (2008) 1693–1709.

[5] Y. Du, S. Li, Positive solutions with prescribed patterns in some simple semilinearequations, Differential Integral Equations 12 (2002) 805–822.

[6] P. Freitas, Nonlocal reaction-diffusion equations, in: Differential Equations with Ap-plications to Biology, Halifax, NS, 1997, in: Fields Inst. Commun., vol. 21, Amer.Math. Soc., Providence, RI, 1999, pp. 187–204.

[7] J. Garcıa-Melian, J.D. Rossi, A logistic equation with refuge and nonlocal diffusion,Commun. Pure Appl. Anal. 8 (2009) 2037–2053.

[8] J. Garcıa-Melian, J.D. Rossi, On the principal eigenvalue of some nonlocal diffusionproblems, J. Differential Equations 246 (2009) 21–38.

[9] V. Hutson, S. Martinez, K. Mischaikow, G.T. Vickers, The evolution of dispersal, J.Math. Biol. 47 (2003) 483–517.

[10] J.W. Sun, W.T. Li, Positive solutions of periodic logistic equation with nonlocaldispersal, preprint, 2013.

[11] J.W. Sun, W.T. Li, Z.C. Wang, A nonlocal dispersal Logistic model with spatialdegeneracy, Discrete Contin. Dyn. Syst., in press.

[12] J.-W. Sun, F.-Y. Yang, W.-T. Li, A nonlocal dispersal equation arising from aselection-migration model in genetics, J. Differential Equations 257 (2014) 1372–1402.

blow-up profiles for positive solutions of nonlocal dispersal equation

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