Dirichlet problem of heat equation for \(C^ 2\) domains
Let \(t=\tau (x)\) be a \(C^ 2\) function on \({\mathbb{R}}\), \(\Omega\) the region \(\{(x,t):\quad t>\tau (x)\}.\) For example the following theorem is proved: Let \(\sup | \tau ''| \equiv A<+\infty\), \(f\in L_ p\), \(1\leq p<\infty\), and \[ u(x,t)=\int_{{\mathbb{R}}}K(x,t;y,\tau (y))f(y)dy,\quad for\quad (x,t)\in \Omega, \] where \[ K(x,t;y,\tau (y))=\frac{1}{2\sqrt{\pi (t-\tau (y))}}e^{-(x-y)^ 2/4(t-\tau (y))}[1- \tau '(y)\frac{x-y}{t-\tau (y)}]\quad for\quad t>\tau (y);\quad 0\quad for\quad t\leq \tau (y). \] Then u(x,t) is a solution of the heat equation in \(\Omega\), such that at almost every boundary point \((x_ 0,\tau (x_ 0))\) the non-tangential limit \(u^*(x_ 0,\tau (x_ 0))\) exists and \[ u^*(x_ 0,\tau (x_ 0))=f(x_ 0)+\int_{{\mathbb{R}}}K(x_ 0,\tau (x_ 0);y,\tau (y))f(y)dy. \]
scientific article; zbMATH DE number 4111175
scientific article