This chapter is devoted to the semigroup approach to a class of initialboundary value problems for semilinear parabolic differential equations. We prove Theorem 1.5 by using the theory of fractional powers of analytic semigroups (Theorems 10.1 and 10.2). To do this, we verify that all the conditions of Theorem 2.8 are satisfied. Our semigroup approach here can be traced back to the pioneering work of Fujita–Kato [FK]. For detailed studies of semilinear parabolic equations, the reader is referred to Friedman [Fr1], Henry [He] and also [Ta4].