On the asymptotics of solutions of elliptic equations at the ends of 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 has an 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 Riemannian manifold 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 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 $(\varphi(\theta),\psi(\theta))$ 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.