Singular behavior of the solution of the Cauchy-Dirichlet heat equation in weighted $L^p$-Sobolev spaces

Abstract

We consider the heat equation on a polygonal domain $\Omega$ of the plane in weighted $L^p$-Sobolev spaces \begin{equation} \label{ab1} \begin{array}{cl} \partial_t u -\Delta u = h, & \mbox{in } \Omega \times {]0,T[}, \\ u=0,& \mbox{on } \partial\Omega \times {[0,T]}, \\ u(\cdot, 0)=0,& \mbox{in } \Omega. \end{array} \end{equation} Here $h$ belongs to $L^p(0,T;L^p_\mu(\Omega))$, where $L^p_\mu(\Omega)=\{v \in L^p_{loc}(\Omega): r^\mu v\in L^p(\Omega)\},$ with a real parameter $\mu$ and $r(x)$ the distance from $x$ to the set of corners of $\Omega$. We give sufficient conditions on $\mu$, $p$ and $\Omega$ that guarantee that problem (\ref{ab1}) has a unique solution $u\in L^p(0,T;L^p_\mu(\Omega))$ that admits a decomposition into a regular part in weighted $L^p$-Sobolev spaces and an explicit singular part.