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

2022/02/14 by Jin Liang, Jun Yan, Jin, Liang +3 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Biology Tumor Growth #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2202.06525

openalex publication_date 2022/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we discuss the existence and multiplicity problem of viscosity solution to the Hamilton-Jacobi equation h(x,d_x u)+λ(x)u=c, x∈ M, where M is a closed manifold and λ:M→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.

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