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.
Citation
Colette De Coster. Serge Nicaise. "Lower and upper solutions for the heat equation on a polygonal domain of $\mathbb R^2$." Differential Integral Equations 26 (5/6) 603 - 622, May/June 2013. https://doi.org/10.57262/die/1363266080
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