On a Fourth Order Parabolic Equation in a Nonregular Domain of ${ \mathbb{R} ^{3}}$

Abstract

In this work, we give new results of existence, uniqueness and maximal regularity of a solution to the following two space linear fourth order parabolic equation ${\partial_{t}u + \partial _{x_{1}}^{4}u + \partial _{x_{2}}^{4}u = f}$ , set in a domain ${Q = \left\{\left( t, x_{1} \right) \in \mathbb{R}^{2} : 0 < t < T; \varphi _{1}\left( t \right) < x_{1} < \varphi _{2}\left(t \right) \right\}\times \left] 0, b \right[ }$ of ${\mathbb{R} ^{3}}$ , with Cauchy-Dirichlet boundary conditions, under some assumptions on the functions (${\varphi _{i}}$) _{i=1, 2}. The right-hand side f of the equation is taken in a weighted Lebesgue space. One of the main issues of this work is that the domain Q can possibly be nonregular, for instance, the singular case where ${\varphi_{1} \left( 0 \right) = \varphi _{2}\left( 0 \right) }$ is allowed. The method used is based on the approximation of the domain Q by a sequence of subdomains ${\left( Q_{n} \right) _{n}}$ which can be transformed into regular domains. This work is an extension of the one space variable case studied in [1].