On generalized principal eigenvalues of nonlocal operators witha drift

Abstract

This article is concerned with the following spectral problem: to find a positive function φ∈C1(Ω) and λ∈R such that q(x)φ′(x)+∫ΩJ(x,y)φ(y)dy+a(x)φ(x)+λφ(x)=0forx∈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 (λp,φp) and derive some of its main properties. In particular, we prove that λp(Ω)=limR→+∞λp(ΩR), where ΩR=Ω∩(−R,R) and λp(ΩR) corresponds to the principal eigenvalue of the truncation operator defined in ΩR. The proofs especially rely on the derivation of a new Harnack type inequality for positive solutions of such problems.