On prescribed change of profile for solutions of parabolic equations
  Parabolic equations with homogeneous Dirichlet conditions on the boundary are studied in a setting where the solutions are required to have a prescribed change of the profile in fixed time, instead of a Cauchy condition. It is shown that this problem is well-posed in L_2-setting. Existence and regularity results are established, as well as an analog of the maximum principle. 
Nikolai Dokuchaev
math.AP