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

Abstract

We consider the heat equation in a polygonal domain $\Omega$ of the plane in weighted $L^p$-Sobolev spaces \begin{equation} \begin{array}{cl} \partial_t u -\Delta u = h, & \mbox{in } \Omega \times {(-\pi,\pi ) }, \\ u=0,& \mbox{on } \partial\Omega \times {[-\pi,\pi]}, \\ u(\cdot, -\pi)=u(\cdot, \pi),& \mbox{in } \Omega. \end{array} \tag*{(0.1)} \end{equation} Here $h$ belongs to $L^p(-\pi,\pi;L^p_\mu(\Omega))$, 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 (0.1) has a unique solution $u\in L^p(-\pi,\pi;L^p_\mu(\Omega))$ that admits a decomposition into a regular part in weighted $L^p$-Sobolev spaces and an explicit singular part. The classical Fourier transform techniques do not allow one to handle such a general case. Hence we use the theory of sums of operators.