Abstract:
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.
Keywords:non-compact Riemannian manifold, end of a manifold, spectral equation,
asymptotic distinguishability, generalized Cauchy problem.
Fulltext:
PDF file (760 kB)
