Existence of solution to parabolic equations with mixed boundary condition on non-cylindrical domains

In this paper we study the existence of weak solutions to initial boundary value problems of linear and semi-linear parabolic equations with mixed boundary conditions on non-cylindrical domains ⋃t∈(0,T)Ω(t)×{t} of spatial-temporal space RN×R. In the case of the linear equation, each boundary condition is given on any open subset of the boundary surface Σ=⋃t∈(0,T)∂Ω(t)×{t} under a condition that the boundary portion for Dirichlet condition Σ0⊂Σ is nonempty at any time t. Due to this, it is difficult to reduce the problem to the one on a cylindrical domain by diffeomorphism of the spatial domains Ω(t). By a transformation of the unknown function and the penalty method, we connect the problem to a monotone operator equation for functions defined on the non-cylindrical domain. We are also concerned with a semilinear problem when the boundary portion for Dirichlet condition is cylindrical.