Singular Parabolic Partial Differential Equations with Time Dependent Coefficients
Doris S. Stockton · Transactions of the American Mathematical Society · 1962
Introduction.In this paper we consider the general operator (0.1) Si(t) = Dllit)Dm + c(x,t) lVjíb:' i = x ^ r2.In (0.1) for each t, p(t) = p, is a Borel measure defined on the Borel sets of {rur2), o(t) = o, is a strictly increasing function continuous on {rur2), c(x,t) ^ 0, and c(x,i) is continuous in x on [r^] for each ts\a,b~\.The generalized derivatives Dff(() and DmDa(t) are defined in §2.The operator Si(t) is, for each t e [a,f>], a linear, but not necessarily bounded, operator.The purpose of this paper is to present conditions on pt and a, such that given an initial value for f = a and possible boundary conditions at rt and r2 there exists a unique solution of the equationt) "Z^Z' rlS*2 r2-Four types of boundaries, regular, entrance, exit, and natural are considered.These types are defined in §2.The boundary classifications of (0.1) for the ot and p, independent of t were formulated by K. Itô and used by H. McKean [11] in accordance with W. Feller [4], and the boundary conditions (2.8) are the generalized classical ones.W. Feller [3; 2] has shown that for each t, Si(t) is a generalization of the classical differential operator(See §2 for the pertinent details of this demonstration.)In (0.3) for each t, a(x,t)>0, a(x,t) and b(x,t) are continuous in x on (rur2), c{x,t) g 0, and c(x,t) is continuous in x on [r1,r2].The operators we consider are allowed to be singular in that a(x,i) and b(x,t) are allowed to have discontinuities in x at the boundaries rt and r2 and fl(x,i) may vanish at the boundaries.The boundary classifications are seen to depend on the behavior of b*(x,t) = b(x,t)/a(x,t) and the boundary con-