Imposing the change of an evolution profile

Is it possible to prescribe the change in the profile of an evolution process over a given interval of time? This question has been recently raised and answered in the affirmative when the evolution process is one of diffusion-convection-reaction described by a uniformly parabolic, second-order, homogeneous, linear equation. The present paper shows that the conclusion holds for even more general diffusion-convection-reaction processes, including those described by inhomogeneous equations, by a class of nonlinear degenerate parabolic equations, and by fully nonlinear uniformly parabolic equations. Further generalizations and limitations are explored.