Nonlinear Semigroup Approach to the Hamilton-Jacobi Equation– a Toy Model
Keywords:
Hamilton-Jacobi equations, viscosity solutions, bifurcation phenomenonAbstract
We discuss the existence and multiplicity problem of viscosity solution to the Hamilton-Jacobi equation h(x,dxu) + 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 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.
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