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.
Adv. Differential Equations
16(3/4):
221-256
(March/April 2011).
DOI: 10.57262/ade/1355854308
ACCESS THE FULL ARTICLE
It is not available for individual sale.
This article is only available to subscribers.
It is not available for individual sale.
It is not available for individual sale.