On generalized principal eigenvalues of nonlocal operators with a drift

This article is concerned with the following spectral problem: to find a positive function phi is an element of C-1 (Omega) and lambda is an element of R such that q(x)phi'(x) + integral(Omega) J(x,y)phi(y) dy + a(x)phi(x) + lambda phi(x) = 0 for x is an element of Omega, where Omega subset of 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 (lambda(p), phi(p)) and derive some of its main properties. In particular, we prove that lambda(p)(Omega) = lim(R ->+infinity) lambda(p)(Omega(R)) where Omega(R) = Omega boolean AND (-R,R) and lambda(p)(Omega(R)) corresponds to the principal eigenvalue of the truncation operator defined in Omega(R). The proofs especially rely on the derivation of a new Harnack type inequality for positive solutions of such problems. (C) 2019 Elsevier Ltd. All rights reserved.