Existence of solution to parabolic equations with mixed boundary   condition on non-cylindrical domain
  In this paper we are concerned with the initial boundary value problems of linear and semi-linear parabolic equations with mixed boundary conditions on non-cylindrical domains in spatial-temporal space. We obtain the existence of a weak solution to the problem. In the case of the linear equation the parts for every type of boundary condition are any open subsets of the boundary being nonempty the part for Dirichlet condition at any time. Due to this it is difficult to reduce the problem to one on a cylindrical domain by diffeomorphism of the domain. By a transformation of unknown function and penalty method we connect the problem to a monotone operator equation for functions defined on the non-cylindrical domain. In this way a semilinear problem is considered when the part of boundary for Dirichlet condition is cylindrical. 
Tujin Kim, Daomin Cao
math.AP