LAUSR

Abstract 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 R N × 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 Σ 0 ⊂ 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.