Parabolic equations with Cauchy–Dirichlet boundary conditions in a non-regular domain of ℝN+1

Abstract

Abstract. New results on the existence, uniqueness and maximal regularity of a solution are given for a parabolic equation set in a non-regular domain Q = ( t , x 1 ) ∈ ℝ 2 : 0 < t < T ; ϕ 1 ( t ) < x 1 < ϕ 2 ( t ) × ∏ i = 1 N - 1 ] 0 , b i [ $ Q=\bigl \lbrace (t,x_1) \in \mathbb {R}^{2}:0<t<T;\; \varphi _1(t) <x_1<\varphi _{2}(t) \bigr \rbrace \times \prod _{i=1}^{N-1}{]0,b_i[} $ of ℝN+1, with Cauchy–Dirichlet boundary conditions, under some assumptions on the functions ϕ 1 $\varphi _1$ and ϕ 2 $\varphi _2$ . The right-hand side term of the equation is taken in L 2 ( Q ) $L^{2}(Q)$ . The method used is based on the approximation of the domain Q by a sequence of sub-domains ( Q n ) $(Q_n)$ which can be transformed into regular domains. This work is an extension of the two space variables case studied in [Differ. Equ. Appl. 2 (2010), no. 2, 251–263].