Elliptic and parabolic equations with Dirichlet conditions at infinity on Riemannian manifolds

TL;DR

This paper studies existence, uniqueness, and asymptotic behavior of solutions to elliptic and parabolic equations with conditions at infinity on noncompact Riemannian manifolds, extending classical PDE results to geometric settings.

Contribution

It establishes new existence and uniqueness results for PDEs with boundary conditions at infinity on Riemannian manifolds, including large-time behavior analysis.

Findings

Existence of solutions with prescribed behavior at infinity.

Uniqueness of solutions under certain geometric conditions.

Analysis of large-time asymptotics of solutions.

Abstract

We investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in $M\times \mathbb R_+$, where $M$ is a complete noncompact Riemannian manifold. Under specific assumptions, we establish existence of solutions satisfying prescribed conditions at infinity, depending on the direction along which infinity is approached. Moreover, the large-time behavior of such solutions is studied. We consider also elliptic equations on $M$ with similar conditions at infinity.