This monograph is devoted to the functional analytic approach to initial boundary value problems for semilinear parabolic differential equations. First we study non-homogeneous boundary value problems for second-order elliptic differential operators, in the framework of Sobolev spaces of Lp style, which include as particular cases the Dirichlet and Neumann problems. We prove that these boundary value problems provide an example of analytic semigroups in the Lp topology. The essential point in the proof is to define a function space which is a tool well suited to investigating our boundary conditions. By virtue of the theory of analytic semigroups, one can apply this result to the study of the initial boundary value problems for semilinear parabolic differential equations in the framework of Lp spaces.

This monograph grew out of a set of lecture notes “On initial boundary value problems for semilinear parabolic differential equations” for graduate courses given at the University of Tsukuba in winter 1994/95. In order to make this monograph accessible to a broad readership, I have tried to start from scratch. In the preparatory chapters, we even prove fundamental results like a generation theorem for analytic semigroups in functional analysis and Sobolev imbeddings theorems in partial differential equations. Furthermore, we summarize the basic definitions and results about the Lp theory of pseudo-differential operators which is considered as a modern theory of potentials. The Lp theory of pseudo-differential operators forms a most convenient tool in the study of elliptic boundary value problems in the framework of Sobolev spaces of Lp style. The material in these preparatory chapters is given for completeness, to minimize the necessity of consulting too many outside references. This makes the monograph fairly self-contained.