Generalized solution of the Hamilton–Jacobi equation with a three-component Hamiltonian

On a bounded time interval, we consider the Cauchy problem for the of evolutionary Hamilton–Jacobi equation in the case where the dimension of the phase variable is equal to one. The Hamiltonian depends on the phase and momentum variables and the dependence on the momentum variable is exponential. The domain in which the equation is considered is divided into three subdomains. Inside each of the three subdomains, the Hamiltonian is continuous, while at the boundaries of these subdomains it is discontinuous with respect to the phase variable. Based on the minimax/viscosity approach, we introduce the notion of a continuous generalized solution of the problem and prove its existence. The generalized solution is unique if the problem is considered in a domain bounded with respect to the phase variable.