Parabolic partial differential equations with discrete state-dependent   delay: classical solutions and solution manifold
  Classical solutions to PDEs with discrete state-dependent delay are studied. We prove the well-posedness in a set $X_F$ which is an analogous to the solution manifold used for ordinary differential equations with state-dependent delay. We prove that the evolution operators are $C^1$-smooth on the solution manifold. 
Tibor Krisztin and Alexander Rezounenko
math.AP