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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:265

Article

Kim, Tujin¹  Cao, Daomin^2,3 

[1] Acad Sci, Inst Math, Pyongyang, North Korea

[2] Chinese Acad Sci, Inst Appl Math, AMSS, Beijing 100190, Peoples R China

[3] Univ Chinese Acad Sci, Beijing 100049, Peoples R China

关键词: Parabolic equation;    Non-cylindrical domain;    Mixed boundary condition;    Existence;   

DOI  :  10.1016/j.jde.2018.04.046

来源: Elsevier

【 摘 要 】

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 U boolean OR(t is an element of(0, T)) Omega(t) x {t} of spatial-temporal space R-N x R. In the case of the linear equation, each boundary condition is given on any open subset of the boundary surface Sigma = U boolean OR(t is an element of(0, T) )partial derivative Omega(t) x {t} under a condition that the boundary portion for Dirichlet condition Sigma(0) subset of Sigma 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 Omega(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. (C) 2018 Published by Elsevier Inc.

【 授权许可】

Free