New Integral Representations for Solutions of Cauchy's Problem for Abstract Parabolic Equations - Donaldson, J. A. (1971)
A study has been made of Cauchy's problem for a class of abstract parabolic differential equations. Our study is based upon a transformation that maps solutions of second-order abstract Cauchy problems into solutions of first-order. This is a preliminary report on some of the results obtained. These results include (1) new representations for solutions of abstract parabolic equations, (2) applications to the heat equation, (3) a discussion of solution representations for second-order abstract Cauchy problems, and (4) applications to nonwell posed (in the sense of Hadamard) Cauchy problems.