Construction of a continuous minimax/viscosity solution of the Hamilton–Jacobi–Bellman equation with nonextendable characteristics

The Cauchy problem for the Hamilton–Jacobi equation, which appears in molecular biology for the Crow–Kimura model of molecular evolution, is considered. The state characteristics of the equation that start in a given initial manifold bounded in the state space stay in a strip bounded in the state variable and fill a part of this strip. The values attained by the impulse characteristics on a finite time interval are arbitrarily large in magnitude. We propose a construction of a smooth extension for a continuous minimax/viscosity solution of the problem to the part of the strip that is not covered by the characteristics starting in the initial manifold.