Asymptotic Behaviors for Nonlocal Diffusion Equations about the Dispersal Spread

Su, Yuan-Hang; Li, Wan–Tong; Yang, Fei-Ying

doi:10.48550/arxiv.1911.07665

2019

DOI: 10.48550/arxiv.1911.07665

|View full text |Cite

Preprint

Asymptotic Behaviors for Nonlocal Diffusion Equations about the Dispersal Spread

Yuan-Hang Su

Wan–Tong Li

Fei-Ying Yang

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2020

Publication Types

Select...

Article1

Relationship

Self Cite1

Independent0

Authors

Journals

Cited by 1 publication

(3 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, where λ r is the principal eigenvalue of the corresponding time-periodic random dispersal eigenvalue problem. For the case m > 2, we conjecture that lim σ→0 + λ p (L τ,µ,σ,m Ω ) = − ā, which has been proved by [38,Theorem 1.3 (iii)] for the time-independent operators. In the second part of this paper, we consider the applications of previous results for the generalised principal eigenvalue to the nonlocal dispersal equation in spatio-temporally heterogeneous environments…”

mentioning

confidence: 61%

“…Nonlocal dispersal evolution equations of the form (1.2) have attracted a lot of attentions in recent years; See [29,31,33,36,40] and references therein. The case f (x, t, u) = f (x, u) in equations (1.2) has been well studied; See [3,5,7,9,12,20,34,35,[37][38][39]42]. We first recall the following results of the existence and non-existence of positive time-periodic solutions to (1.3) by Rawal and Shen [29] and Shen and Vo [36]: Lemma 1.6.…”

mentioning

confidence: 99%

“…is the unique positive 1-periodic solution of (1.4). We refer interested readers to [38,Theorem 1.8 (iii)] for the time-independent case.…”

mentioning

confidence: 99%

See 2 more Smart Citations

The generalised principal eigenvalue of time-periodic nonlocal dispersal operators and applications

Lou

et al. 2020

Journal of Differential Equations

Self Cite

View full text Add to dashboard Cite

This paper is mainly concerned with the generalised principal eigenvalue for timeperiodic nonlocal dispersal operators. We first establish the equivalence between two different characterisations of the generalised principal eigenvalue. We further investigate the dependence of the generalised principal eigenvalue on the frequency, the dispersal rate and the dispersal spread. Finally, these qualitative results for time-periodic linear operators are applied to time-periodic nonlinear KPP equations with nonlocal dispersal, focusing on the effects of the frequency, the dispersal rate and the dispersal spread on the existence and stability of positive time-periodic solutions to nonlinear equations.where (x, t) ∈Ω× R, Ω ⊂ R N is a bounded domain, τ > 0 is the frequency, µ > 0 is the dispersal rate, σ > 0 is the dispersal spread which characterises the dispersal range, m ≥ 0 is the cost parameter, J σ (·) = 1 σ N J( · σ ) is the scaled dispersal kernel. Throughout the paper, we will make the following assumptions on the dispersal kernel J, a family of functions {h σ } σ>0 and function a:Define the spaces χ Ω , χ + Ω , χ ++ Ω as follows:where C 0,1 (Ω × R) denotes the class of functions that are continuous in x and C 1 in t. The operator L τ,µ,σ,m Ω is then considered as an unbounded linear operator on the space C 1 (Ω × R) with domain χ Ω , namely,It may be worthwhile to point out that the time-periodic nonlocal dispersal operators of the form (1.1) include several kinds of boundary conditions, such as Dirichlet, Neumann and mixed type; See [7,28,29].The principal eigenvalues for time-periodic nonlocal dispersal operators have been studied in [2, 19, 29-31, 33, 36, 40]. In this paper, we adopt the approaches as in Berestycki, Nirenberg and Varadhan [6] and Berestycki, Coville and Vo [4] for the definition of the generalised principal eigenvalue λ p (L τ,µ,σ,m Ω ):TIME PERIODIC NONLOCAL DISPERSAL OPERATORS AND APPLICATIONS 3 Another definition for the generalised principal eigenvalue of L τ,µ,σ,m Ω is given by λ ′ p (L τ,µ,σ,m Ω ) := inf{λ ∈ R | ∃ v ∈ χ ++ Ω s.t. (L τ,µ,σ,m Ω + λ)[v] ≥ 0 inΩ × R}, motivated by the works of Donsker and Varadhan in [13] and Berestycki, Coville and Vo in [4].Shen and Vo proved in [36] that λ p = λ ′ p when λ p is the principal eigenvalue and h σ (x) ≡ 1 for all σ > 0. Our first main result proves that for general h σ , λ p = λ ′ p always holds, and λ p can be characterised as the infimum of the spectrum of −L τ,µ,σ,m Ω , whether λ p is an eigenvalue or not. Theorem 1.1. Assume that (J), (H) and (A) hold. Then λ p (L τ,µ,σ,m Ω ) = λ ′ p (L τ,µ,σ,m Ω ) = λ 1 , where λ 1 = inf{Reλ | λ ∈ σ(−L τ,µ,σ,m Ω )} and σ(−L τ,µ,σ,m Ω ) is the spectrum of −L τ,µ,σ,m Ω . Next, we turn to study the influences of the frequency τ , the dispersal rate µ and the dispersal spread σ on the generalised principal eigenvalue λ p (L τ,µ,σ,m Ω ). The following result establishes the monotonicity and asymptotic behaviors of the generalised principal eigenvalue λ p (L τ,µ,σ,m Ω ) with respect to the frequency τ : Theorem...

show abstract

mentioning

confidence: 61%

mentioning

confidence: 99%

See 1 more Smart Citation

The generalised principal eigenvalue of time-periodic nonlocal dispersal operators and applications

Lou

et al. 2020

Journal of Differential Equations

Self Cite

View full text Add to dashboard Cite

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Asymptotic Behaviors for Nonlocal Diffusion Equations about the Dispersal Spread

Cited by 1 publication

References 36 publications

The generalised principal eigenvalue of time-periodic nonlocal dispersal operators and applications

The generalised principal eigenvalue of time-periodic nonlocal dispersal operators and applications

Contact Info

Product

Resources

About