A nonlinear semigroup approach to Hamilton-Jacobi equations--revisited

TL;DR

This paper revisits Hamilton-Jacobi equations on Riemannian manifolds, analyzing the structure of viscosity solutions and their long-term behavior under more general conditions than previous studies.

Contribution

It extends prior work by Jin-Yan-Zhao to include detailed viscosity solution structures and large time behavior analysis without Tonelli conditions.

Findings

Detailed structure of viscosity solutions characterized.

Large time behavior of solutions analyzed.

Extension beyond Tonelli conditions achieved.

Abstract

We consider the Hamilton-Jacobi equation \[{H}(x,Du)+\lambda(x)u=c,\quad x\in M, \] where $M$ is a connected, closed and smooth Riemannian manifold. The functions ${H}(x,p)$ and $\lambda(x)$ are continuous. ${H}(x,p)$ is convex, coercive with respect to $p$, and $\lambda(x)$ changes the signs. The first breakthrough to this model was achieved by Jin-Yan-Zhao \cite{JYZ} under the Tonelli conditions. In this paper, we consider more detailed structure of the viscosity solution set and large time behavior of the viscosity solution on the Cauchy problem.