On generalized principal eigenvalues of nonlocal operators witha drift. (April 2020) - Coville, Jérôme; Hamel, François
Abstract: 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 ) + ∫ Ω J ( x, y ) φ ( y ) d y + a ( x ) φ ( x ) + λ φ ( 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.