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On some boundary value problems for the heat equation in a non-regular type of a prism of ℝ N+1

On some boundary value problems for the heat equation in a non-regular type of a prism of ℝ N+1 AbstractThis paper is devoted to the analysis of the boundary value problem∂t⁡u-Δ⁢u=f{\partial_{t}u-\Delta u=f},with an N-dimensional space variable, subject to a Dirichlet–Robin type boundary condition on the lateral boundary of the domain.The problem is settled in a noncylindrical domain of the form Q={(t,x1)∈ℝ2:0<t<T,φ1(t)<x1<φ2(t)}×∏i=1N-1]0,bi[Q=\{(t,x_{1})\in\mathbb{R}^{2}:0<t<T,\varphi_{1}(t)<x_{1}<\varphi_{2}(t)\}%\times\prod_{i=1}^{N-1}{]0,b_{i}[}, where φ1{\varphi_{1}}and φ2{\varphi_{2}}are smooth functions.One of the main issues of the paper is that the domain can possibly be non-regular; for instance, the significant case when φ1⁢(0)=φ2⁢(0){\varphi_{1}(0)=\varphi_{2}(0)}is allowed. We prove well-posedness results for the problem in a number of different settings and under natural assumptions on the coefficients and on the geometrical properties of the domain.This work is an extension of the one-dimensional case studied in [4]. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Georgian Mathematical Journal de Gruyter

On some boundary value problems for the heat equation in a non-regular type of a prism of ℝ N+1

Georgian Mathematical Journal , Volume 25 (3): 13 – Sep 1, 2018

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Publisher
de Gruyter
Copyright
© 2018 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1572-9176
eISSN
1572-9176
DOI
10.1515/gmj-2017-0053
Publisher site
See Article on Publisher Site

Abstract

AbstractThis paper is devoted to the analysis of the boundary value problem∂t⁡u-Δ⁢u=f{\partial_{t}u-\Delta u=f},with an N-dimensional space variable, subject to a Dirichlet–Robin type boundary condition on the lateral boundary of the domain.The problem is settled in a noncylindrical domain of the form Q={(t,x1)∈ℝ2:0<t<T,φ1(t)<x1<φ2(t)}×∏i=1N-1]0,bi[Q=\{(t,x_{1})\in\mathbb{R}^{2}:0<t<T,\varphi_{1}(t)<x_{1}<\varphi_{2}(t)\}%\times\prod_{i=1}^{N-1}{]0,b_{i}[}, where φ1{\varphi_{1}}and φ2{\varphi_{2}}are smooth functions.One of the main issues of the paper is that the domain can possibly be non-regular; for instance, the significant case when φ1⁢(0)=φ2⁢(0){\varphi_{1}(0)=\varphi_{2}(0)}is allowed. We prove well-posedness results for the problem in a number of different settings and under natural assumptions on the coefficients and on the geometrical properties of the domain.This work is an extension of the one-dimensional case studied in [4].

Journal

Georgian Mathematical Journalde Gruyter

Published: Sep 1, 2018

References