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
jl@njust.edu.cn
Jun Yan
School of Mathematical Sciences, Fudan University, Shanghai, China
yanjun@fudan.edu.cn
Kai Zhao
School of Mathematical Sciences, Fudan University, Shanghai, China
zhao_kai@fudan.edu.cn
[Abstract-pdf]
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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