Well-posedness for constrained Hamilton-Jacobi equations

The goal of this paper is to study a Hamilton-Jacobi equation \begin{equation*} \begin{cases} u_t=H(Du)+R(x,I(t)) &$\text{in }$$\mathbb{R}$^n \times (0,\infty), \sup_{$\mathbb{R}$^n} u(\cdot,t)=0 &$\text{on }$[0,\infty), \end{cases} \end{equation*} with initial conditions $I(0)=0$, $u_0(x,0)=u_0(x)$ on ${\mathbb{R}}^n$. Here $(u,I)$ is a pair of unknowns and the Hamiltonian $H$ and the reaction $R$ are given. And $I(t)$ is an unknown constraint (Lagrange multiplier) that forces supremum of $u$ to be always zero. We construct a solution in the viscosity setting using a fixed point argument when the reaction term $R(x,I)$ is strictly decreasing in $I$. We also discuss both uniqueness and nonuniqueness. For uniqueness, a certain structural assumption on $R(x,I)$ is needed. We also provide an example with infinitely many solutions when the reaction term is not strictly decreasing in $I$.