On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators

By Jérôme Coville. 2010

229 citations recorded by the source corpus.

Abstract In this paper we are interested in the existence of a principal eigenfunction of a nonlocal operator which appears in the description of various phenomena ranging from population dynamics to micro-magnetism. More precisely, we study the following eigenvalue problem: ∫ Ω J ( x − y g ( y ) ) ϕ ( y ) g n ( y ) d y + a ( x ) ϕ = ρ ϕ , where Ω ⊂ R n is an open connected set, J a non-negative kernel and g a positive function. First, we establish a criterion for the existence of a principal eigenpair ( λ p , ϕ p ) . We also explore the relation between the sign of the largest element of the spectrum with a strong maximum property satisfied by the operator. As an application of these results we construct and characterise the solutions of some nonlinear nonlocal reaction diffusion equations.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/a492daf8848f39b0ce67783d6b251ba310b047f6

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Citations in the corpus

The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries (2018, background): “, [5, 11, 29]) that λp(L(l1,l2) + a0) is a principal eigenvalue.”
Reaction-diffusion-advection models for the effects and evolution of dispersal (2013, background): “) Models similar to (44) and related models for interacting species have been studied by various researchers; see [4, 20, 26, 27, 28, 29, 47, 53, 55] and the references in those papers.”
Two species nonlocal diffusion systems with free boundaries (2019, background): “Therefore there exists L ≥ L0 such that λp ( L(−l,l) + al ) > 0 for l > L, and by [3, 7] and Proposition 3.”
On Principal Spectrum Points/Principal Eigenvalues of Nonlocal Dispersal Operators and Applications (2013, background): “It is known that a nonlocal dispersal operator may not have a principal eigenvalue (see [9], [34] for examples), which reveals some essential difference between nonlocal and random dispersal operators.”
Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats (2012, methodology): “We learned about the work [10] when the current paper was almost finished.”
Criteria for the Existence and Lower Bounds of Principal Eigenvalues of Time Periodic Nonlocal Dispersal Operators and Applications (2012, background): “The principal eigenvalue problem for time independent nonlocal dispersal operators with Dirichlet type, or Neumann type, or periodic boundary condition has been recently studied by many people (see [9,15,18,22,31,30], and references therein) and is quite well understood now.”
Pulsating fronts for nonlocal dispersion and KPP nonlinearity (2013, unknown): “[20] J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Ann.”
Traveling Wave Solutions of Spatially Periodic Nonlocal Monostable Equations (2012, background): “See, for example, [2], [6], [10], [12], [14], [19], [20], [25], [26], [30], [31], [32], [34], [53], for the study of spectral theory for nonlocal dispersal operators and the existence, uniqueness, and stability of nontrivial positive stationary solutions.”
Approximations of random dispersal operators/equations by nonlocal dispersal operators/equations (2014, background): “11) have also been studied recently by many people (see [5, 10, 25, 32, 34, 35, 36, 37], etc.”
On the definition and the properties of the principal eigenvalue of some nonlocal operators (2015, background): “1 Motivations: nonlocal reaction diffusion equation Our interest in studying the properties of λp(LΩ + a) stems from the recent studies of populations having a long range dispersal strategy [25, 28, 4, 41, 60].”
On eigenvalue problems arising from nonlocal diffusion models (2016, background): “Hence the necessary and sufficient condition in [7] is different: λp ≥ 0.”