On the Uniqueness of Solutions to One-Dimensional Constrained Hamilton-Jacobi Equations

The goal of this paper is to study the uniqueness of solutions to a constrained Hamilton-Jacobi equation { u_t = u²x + R(x, I(t)) in R × (0, ∞), max_R u(·, t) = 0 on [0, ∞), with an initial condition u(x, 0) = u₀ (x) on R. A reaction term R(x, I(t)) is given while I(t) is an unknown constraint (Lagrange multiplier) that forces maximum of u to be always zero. In the paper, we prove uniqueness of a pair of unknowns (u, I) using dynamic programming principle for a particular class of non-separable reaction R(x, I(t)) when the space is one-dimensional.