Singular parabolic partial differential equations with time dependent coefficients

Abstract

(0.1) Q(t) = D,D(t)Dr(t) + c(x,t) a ?t ? b, 19(t) ~~~~r1 ? x ? r2 In (0.1) for each t, ji(t) = pt is a Borel measure defined on the Borel sets of (r1,r2), v(t) = at is a strictly increasing function continuous on (rl,r2), c(x,t) < 0, and c(x,t) is continuous in x on [r1,r2] for each te-[a,b]. The generalized derivatives Da(t) and D(t)Da(t) are defined in ?2. The operator Q(t) is, for each te [a,b], a linear, but not necessarily bounded, operator. The purpose of this paper is to present conditions on pt and at such that given an initial value for t = a and possible boundary conditions at r1 and r2 there exists a unique solution of the equation