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

Abstract

This paper is devoted to the analysis of the boundary value problem {\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,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 {\varphi_{1}} and {\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 {\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].