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Minimax Theory and its Applications 08 (2023), No. 1, 061--084
Copyright Heldermann Verlag 2023

Nonlinear Semigroup Approach to the Hamilton-Jacobi Equation -- a Toy Model

Liang Jin
School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing, China

Jun Yan
School of Mathematical Sciences, Fudan University, Shanghai, China

Kai Zhao
School of Mathematical Sciences, Fudan University, Shanghai, China


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 the parameter $c$ strides over some 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.

Keywords: Hamilton-Jacobi equations, viscosity solutions, bifurcation phenomenon.

MSC: 35F21,35D40,35A02,35B32.

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