Lower and upper solutions for the heat equation on a polygonal domain of $\mathbb R^2$

Abstract

We consider the nonlinear periodic-Dirichlet heat equation on a polygonal domain $\Omega$ of the plane in weighted $L^p$-Sobolev spaces \begin{align} \partial_t u -\Delta u = f(x,t,u), is & \mbox{in } \Omega \times {(-\pi,\pi)}, \notag \\ u=0,is & \mbox{on } \partial\Omega \times {(-\pi,\pi)}, \tag*{(0.1)} \\ u(\cdot, -\pi)=u(\cdot, \pi)is & \mbox{in } \Omega. \notag \end{align} Here $f$ is $L^p(0,T;L^p_\mu(\Omega))$-Carath\'eodory, 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 prove some existence results of this problem in presence of lower and upper solutions well-ordered or not. We first give existence results in an abstract setting obtained using degree theory. We secondly apply them for polygonal domains of the plane under geometrical constraints.