Linear parabolic equations in regions with re-entrant edges
Ali AZZAM, Erwin Kreyszig · Hokkaido Mathematical Journal · 1982
In a recent paper [1] we studied solutions of the parabolic equation ( 1)Lu(x, t)=a_{ik}(x)u_{x_{i}x_{k}}+a_{i}(x, t)u_{x_{i}}+a(x, t)u-u_{t}=f(x, t) , x=(X_{1}^{ },\cdots, x_{n}) , in a simply connected, bounded region \Omega=G\cross J\subset R^{n+1} , n\geqq 2 , J=\{t|0<t\leqq T\} , satisfying the conditionsu|_{\partial G\cross\overline{J}}=\phi(x, t)j under the following assumptions.(A) a_{ik}\in C^{a}(\overline{G}) , a_{i} , a, f\in C^{\alpha}(\overline{\Omega}) , 0<\alpha<1-, (B) \phi(x, 0)=0 , \phi\in C^{2+\alpha}[\partial G\backslash E)\cross\overline{J}]\cap C^{0}(\partial G\cross\overline{J}) , (C) \omega(P)<\pi for all P\in E .Here E=\cup E_{i} , where E_{1} , \cdots , E_{m} are (n-2) -dimensional edges (the intersec-