A nonlocal nonlinear boundary value problem for the heat equations
Let \(\Omega\subset\mathbb{R}^n\) (\(n=2\) or 3) be a bounded smooth domain with boundary \(\partial\Omega=\Gamma_1\cup\Gamma_2\), where \(\Gamma_1\cap\Gamma_2=\emptyset\). We consider the following mixed initial-boundary value problem for the heat equation: \[ u'-\Delta u=0\quad\text{in }\Omega\times(0,T), \] \[ u=0\quad\text{in }\Gamma_1\times(0,T),\;u=H\Biggl(\int_{\Gamma_2}{\partial u\over\partial\nu} dS\Biggr)\quad\text{on }\Gamma_2\times(0,T),\;u(0)=u^0\quad\text{on }\Omega\times\{0\}, \] where \(H:{\mathcal D}_H\subset\mathbb{R}\to\mathbb{R}\) is a strictly decreasing surjective function, \(H^{-1}\) grows linearly, and \(\nu\) stands for the outward unit normal vector at the boundary. We use the Galerkin method to prove the following existence and uniqueness result. Theorem. For any fixed initial data \(u^0\in L^2(\Omega)\), there exists a unique weak solution \[ u\in L^2(0,T;V)\cap W^{1,2}(0,T;V')\subset C(0,T;L^2(\Omega)), \] where \(V=\{v\in H^1(\Omega)\mid v|_{\Gamma_1}=0\), \(v|_{\Gamma_2}=\) constant (depending on \(v\))\}, and \(V'\) stands for the dual of \(V\).
scientific article; zbMATH DE number 940907