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By Kiyoshi ASANOWe give an operator theoretical treatment of initial value problems for semi-linear parabolic partial differential equations : existence of solutions, uniqueness and regularity. We make use of the theory of fractional powers of operators, the theory of semigroups of operators and L^-estimates for elliptic boundary value problems. § 0. Introduction The purpose of this paper is to derive some results on the initial value problems for semi-linear parabolic partial differential equations. The equation is as follows :Here G is a bounded domain in Euclidean ^-space E n , A is an elliptic partial differential operator on G of order 2m and {Bj}^ is a system of m differential operators on the boundary of G. A and {Bj} satisfies some algebraic conditions ((J?) and (C) in § 3). F may contain the derivatives D%u of u of order less than 2m in its variables.If

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