On generalized solutions for two Hamilton–Jacobi equations with state constraints

Two Cauchy problems for Hamilton-Jacobi equation of the evolutionary type with state constraints are considered on a bounded time interval. The state space is one-dimensional. Hamiltonians of the considered problems depend on the state and momentum variables, and the dependence on the momentum variable is exponential. In the first problem, the Hamiltonian is convex in the momentum variable, and in the second problem, the Hamiltonian is concave in this variable. For the first problem, it is proved that a unique continuous viscosity solution exists, and a scheme is proposed for constructing this solution. The proposed scheme is based on the method of generalized characteristics. For the second problem, it is shown that a continuous viscosity solution does not exist, and to define a generalized solution it is necessary to specify some additional conditions.