$C^1$-smoothness of Nemytskii operators on Sobolev-type spaces of   periodic functions
  We consider a class of Nemytskii superposition operators that covers the nonlinear part of traveling wave models from laser dynamics, population dynamics, and chemical kinetics. Our main result is the $C^1$-continuity property of these operators over Sobolev-type spaces of periodic functions. 
Irina Kmit
math.FA