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

TL;DR

This paper proves the existence of weak solutions for linear and semi-linear parabolic equations with mixed boundary conditions on complex non-cylindrical domains, using transformations and penalty methods.

Contribution

It introduces a novel approach to handle mixed boundary conditions on non-cylindrical domains without domain reduction, establishing existence results for such problems.

Findings

Existence of weak solutions for linear parabolic equations with mixed boundary conditions.

Extension to semi-linear problems with cylindrical Dirichlet boundary parts.

Application of transformation and penalty methods to non-cylindrical domains.

Abstract

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.