On Some Singular Boundary Value Problem for Heat Operator

Abstract

In [2], S. Itô treated the following initial-boundary value problem for heat operator:

$${{(I.B.P.)}_{0}}\left\{ {\begin{array}{*{20}{c}} {(1) \tfrac{\partial }{{\partial t}}u = \Delta u, in \Omega \subset {{R}^{n}},} \hfill \\ {\begin{array}{*{20}{c}} {(2) Bu = a(x)\tfrac{{\partial u}}{{\partial n}} + b(x)u = 0,} & {x \in \partial \Omega ,} \\ \end{array} } \hfill \\ { where \tfrac{{\partial u}}{{\partial n}}is the derivative in the direction of} \hfill \\ { outernormal, and a(x) \geqslant 0, b(x) \geqslant 0, a(x) + b(x) = 1,} \hfill \\ {(3) u{{|}_{{t = 0}}} = {{u}_{0}}(x).} \hfill \\ \end{array} } \right.$$