Degenerate Parabolic Initial-Boundary Value Problems*

in Hilbert space and their realizations in function spaces as initial-boundary value problems for partial differential equations which may contain degenerate or singular coefficients. The Cauchy problem consists of solving (1.1) subject to the initial condition Jdu(0) = h. We are concerned with the case where the solution is given by an analytic semigroup; it is this sense in which the Canchy problem is parabolic. Sufficient conditions for this to be the case are given in Theorem 1; this is a refinement of previously known results [15] to the linear problem and it extends the related work [13] to the (possibly) degenerate situation under consideration. Specifically, we do not assume ~ is invertible, but only that it is symmetric and non-negative. Our primary motivation for considering the Cauchy problem for (1.1) is to show that certain classes of mixed initial-boundary value problems for partial differential equations are well-posed. Theorem 2 shows that if the operators ~ / and 5fl have additional structure which is typical of those operators arising from (possibly degenerate) parabolic problems then the evolution equation (1.1) is equivalent to a partial differential equation