On a solution to the Cauchy problem for the Hamilton–Jacobi equation with state constraints

The Cauchy problem for the Hamilton–Jacobi equation, which appears in molecular biology for the Crow–Kimura model of molecular evolution, is considered. The notion of a continuous generalized solution of this problem with state constraints is introduced on the basis of the viscosity (and/or minimax) approach. A construction of the generalized solution of the problem is proposed, which uses the value function in an auxiliary optimal control problem with a given target set. It is shown that the generalized solution in the considered problem is not unique. The research is based on the generalized method of characteristics for the Hamilton–Jacobi equations in the Dirichlet problem.