Continuous generalized solution of the Hamilton–Jacobi equation with a noncoercive Hamiltonian

In this paper, we consider the Cauchy problem for the Hamilton–Jacobi equation with phase constraints arising in molecular biology. The problem has no classical solutions, and the Hamiltonian does not satisfy the conditions guaranteeing the existence of minimax or viscous generalized solutions. A new continuous generalized solution is obtained. We present sufficient conditions for the existence of a global solution preserving the structure given by the initial manifold. The behavior of such solutions is examined for large values of time.