Criteria for the Existence and Lower Bounds of Principal Eigenvalues of Time Periodic Nonlocal Dispersal Operators and Applications

Rawal, Nar; Shen, Wenxian

doi:10.1007/s10884-012-9276-z

Cited by 94 publications

(101 citation statements)

References 23 publications

Supporting

Mentioning

100

Contrasting

Order By: Relevance

“…As it is known from [1,2,22,24] and references therein, the principal spectral theory of the linear operator associated to the equation (1.1) linearized at zero, namely, the nonlocal parabolic-type operator…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Nonlocal dispersal equations in time-periodic media: Principal spectral theory, limiting properties and long-time dynamics

Shen

2019

Journal of Differential Equations

View full text Add to dashboard Cite

Section: Introduction and Main Resultsmentioning

confidence: 99%

Nonlocal dispersal equations in time-periodic media: Principal spectral theory, limiting properties and long-time dynamics

Shen

2019

Journal of Differential Equations

View full text Add to dashboard Cite

“…This completes the proof of Theorem 1.3.Next, we recall the following lemma in[29, Corollary D]. Assume that (J) and(A) hold.…”

mentioning

confidence: 61%

“…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. Assume that (J) and (F) hold.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

“…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].…”

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

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

Principal spectral theory of time‐periodic nonlocal dispersal operators of Neumann type

2022

Mathematische Nachrichten

View full text Add to dashboard Cite

In this communication, we prove some limits of the principal eigenvalue for nonlocal operator of Neumann type with respect to the parameters, which are significant in the understanding of dynamics of biological populations. We obtain a complete picture about limits of the principal eigenvalue in term of the large and small dispersal rate and dispersal range classified by "Ecological Stable Strategy" of persistence. This solves some open problems remainning in the series of work [3,32,30], in which we have to overcome the new difficulties comparing to [3,32,30] since principal eigenvalue of a nonlocal Neumann operator is not monotone with respect to the domain. The maximum principle for this type of operator is also achieved in this paper. K E Y W O R D S generalized principal eigenvalue, maximum principle, nonlocal Neumann operator, principal eigenvalue M S C ( 2 0 2 0 ) 35B50, 35J60, 47G20

show abstract

Criteria for the Existence and Lower Bounds of Principal Eigenvalues of Time Periodic Nonlocal Dispersal Operators and Applications

Cited by 94 publications

References 23 publications

Nonlocal dispersal equations in time-periodic media: Principal spectral theory, limiting properties and long-time dynamics

Nonlocal dispersal equations in time-periodic media: Principal spectral theory, limiting properties and long-time dynamics

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

Principal spectral theory of time‐periodic nonlocal dispersal operators of Neumann type

Contact Info

Product

Resources

About