Uniqueness and existence of viscosity solutions under a degenerate dynamic boundary condition

We consider the initial boundary value problem for a fully-nonlinear parabolic equation in a half space. The boundary condition we study is a degenerate one in the sense that it does not depend on the normal derivative on the boundary. A typical example is a stationary boundary condition prescribing the value of the time derivative of the unknown function. Our setting also covers the classical Dirichlet boundary condition. We establish a comparison principle for a viscosity sub- and supersolution under a weak continuity assumption on the solutions on the boundary. We also prove existence of solutions and give some examples of solutions under several boundary conditions. We show among other things that, in the sense of viscosity solutions, the stationary boundary condition can be different from the Dirichlet boundary condition which is obtained by integrating the stationary condition