On singular initial-boundary value problems for second order parabolic equations (Q1064465)

The author considers the following problem, in which \(\Omega =\underset \tilde{} R^{n-1}\times]0,\infty [\) and \(n\geq 2:\) Find a solution u of the heat equation on \(\Omega\) \(\times]0,T[\), subject to the initial condition that \(u=u_ 0\) when \(t=0\), and the boundary condition \(a(t,x)\partial u/\partial \nu +b(t,x)u=0\) on \(\partial \Omega \times ]0,T[\), where \[ \partial u/\partial \nu =-\partial u/\partial y+\sum^{n-1}_{j=1}c_ j(t,x)\partial u/\partial x_ i|_{y=0}. \] The main purpose is to prove existence, uniqueness, and regularity of the solution, under the condition that a(t,x)\(\geq 0\). Thus earlier results of \textit{S. Ito} [Jap. J. Math. 27, 55-102 (1958; Zbl 0092.311)] are generalized.