This paper examines generalized solutions to the Hamilton Jacobi equation. The Hamiltonian H(t, x, p) is assumed convex in p but is not constrained to have linear growth in this variable. This corresponds to a certain class of generalized Bolza problems, related to optimal control. Lower semicontinuous solutions are considered and it is shown that there is a unique solution, the so-called value function of the underlying Bolza problem. In proving the main result we use recent improved necessary optimality conditions. Viability is also used in a new way, in connection to differential inclusions with unbounded images.