Differential and Integral Equations

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

Colette De Coster and Serge Nicaise

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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.

Article information

Differential Integral Equations, Volume 26, Number 5/6 (2013), 603-622.

First available in Project Euclid: 14 March 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K55: Nonlinear parabolic equations 35B65: Smoothness and regularity of solutions


De Coster, Colette; Nicaise, Serge. Lower and upper solutions for the heat equation on a polygonal domain of $\mathbb R^2$. Differential Integral Equations 26 (2013), no. 5/6, 603--622. https://projecteuclid.org/euclid.die/1363266080

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