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

We consider a linear elliptic differential equation $\Delta u+c(x)u=0$ defined on a Riemannian manifold $\mathcal{M}$ that hasan end $\mathcal{X}$ on which the metric takes the form$dl^2=h^2(r) dr^2+q^2(r) d\theta^2$ in appropriate coordinates.Here $r\in [r_0,+\infty)$, $\theta\in S$, and $S$ is a smooth compact Riemannianmanifold with metric $d\theta^2$. At the end $\mathcal{X}$, the coefficient$c(x)$ takes the form $c(x)=c(r)$. For ends of parabolic type with suchmetrics, we describe the property of asymptotic distinguishabilityof solutions of this equation. For ends of hyperbolic type, we prove a theoremon the admissible rate of convergence to zero for a difference of solutionsof this equation. For both types of ends, we formulate versions of thegeneralized Cauchy problem with initial data $(\varphi(\theta),\psi(\theta))$at the infinitely remote point and study its solubility. The results obtainedare new and, in the case of ends of parabolic type, somewhat unexpected.

non-compact Riemannian manifold, end of a manifold, spectral equation, asymptotic distinguishability, generalized Cauchy problem