On the asymptotics of solutions of elliptic equations at the ends of non-compact Riemannian manifolds with metrics of a special form

Abstract

We consider a linear elliptic differential equation defined on a Riemannian manifold that has an end on which the metric takes the form in appropriate coordinates. Here , , and is a smooth compact Riemannian manifold with metric . At the end , the coefficient takes the form . For ends of parabolic type with such metrics, we describe the property of asymptotic distinguishability of solutions of this equation. For ends of hyperbolic type, we prove a theorem on the admissible rate of convergence to zero for a difference of solutions of this equation. For both types of ends, we formulate versions of the generalized Cauchy problem with initial data at the infinitely remote point and study its solubility. The results obtained are new and, in the case of ends of parabolic type, somewhat unexpected.