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 secondorder elliptic differential operators, in the framework of Sobolev spaces of Lp type, 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, we 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 type. 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.

This work was begun at the University of Turin and the University of Bologna in May 1988 under the sponsorship of the Italian “Consiglio Nazionale delle Ricerche” and a major part of the work was done at the University of the Philippines in the course of the JSPS–DOST exchange program from January l989 to March 1989 while I was on leave from the University of Tsukuba. I take this opportunity to express my gratitude to all these institutions for their hospitality and support.

Thanks are also due to the editorial staff of Cambridge University Press for their unfailing helpfulness and cooperation during the production of the book.