Nonlinear semigroup approach to Hamilton-Jacobi equations -- A toy model

TL;DR

This paper investigates the existence and multiplicity of viscosity solutions to a Hamilton-Jacobi equation on a closed manifold with sign-changing coefficients, revealing bifurcation phenomena using nonlinear semigroup methods.

Contribution

It introduces a novel nonlinear semigroup approach to analyze solution multiplicity and bifurcations in Hamilton-Jacobi equations with sign-changing coefficients.

Findings

Bifurcation occurs when parameter c crosses a critical value.

Detailed analysis of viscosity solutions for a one-dimensional example.

Method applicable to equations with sign-changing coefficients.

Abstract

In this paper, we discuss the existence and multiplicity problem of viscosity solution to the Hamilton-Jacobi equation $$h(x,d_x u)+\lambda(x)u=c,\quad x\in M,$$ where $M$ is a closed manifold and $\lambda:M\rightarrow\mathbb{R}$ changes signs on $M$, via nonlinear semigroup method. It turns out that a bifurcation phenomenon occurs when parameter $c$ strides over the critical value. As an application of the main result, we analyse the structure of the set of viscosity solutions of an one-dimensional example in detail.