On generalized principal eigenvalues of nonlocal operators with a drift *

Jérôme Coville (1) , François Hamel (2)

1 BioSP - Biostatistique et Processus Spatiaux
2 I2M - Institut de Mathématiques de Marseille

This article is concerned with the following spectral problem: to find a positive function ϕ ∈ C 1 (Ω) and λ ∈ R such that q(x)ϕ (x) + ˆ Ω J(x, y)ϕ(y) dy + a(x)ϕ(x) + λϕ(x) = 0 for x ∈ Ω, where Ω ⊂ R is a non-empty domain (open interval), possibly unbounded, J is a positive continuous kernel, and a and q are continuous coefficients. Such a spectral problem naturally arises in the study of nonlocal population dynamics models defined in a space-time varying environment encoding the influence of a climate change through a spatial shift of the coefficient. In such models, working directly in a moving frame that matches the spatial shift leads to consider a problem where the dispersal of the population is modeled by a nonlocal operator with a drift term. Assuming that the drift q is a positive function, for rather general assumptions on J and a, we prove the existence of a principal eigenpair (λ p , ϕ p) and derive some of its main properties. In particular, we prove that λ p (Ω) = lim R→+∞ λ p (Ω R), where Ω R = Ω ∩ (−R, R) and λ p (Ω R) corresponds to the principal eigenvalue of the truncation operator defined in Ω R. The proofs especially rely on the derivation of a new Harnack type inequality for positive solutions of such problems.

Domaines: Equations aux dérivées partielles [math.AP]

Jérôme Coville, François Hamel. On generalized principal eigenvalues of nonlocal operators with a drift *. Nonlinear Analysis: Theory, Methods and Applications, 2019, 193, pp.111569. ⟨10.1016/j.na.2019.07.002⟩. ⟨hal-01966709⟩