This monograph provides a careful and accessible exposition of a functional analytic approach to initial boundary value problems for semilinear parabolic differential equations of second order. It focuses on the relationship between three interrelated subjects in analysis: elliptic boundary value problems and parabolic initial boundary value problems, with emphasis on the general study of analytic semigroups. This semigroup approach can be traced back to the pioneering work of Fujita–Kato [18] on the Navier–Stokes equation in fluid mechanics.

The approach here is distinguished by the extensive use of the techniques characteristic of recent developments in the theory of partial differential equations. The main technique used is the Lp theory of pseudo-differential operators which may be considered as a modern theory of classical potentials. The theory of pseudo-differential operators continues to be one of the most influential works in modern history of analysis, and is a very refined mathematical tool whose full power is yet to be exploited. Several recent developments in the theory of pseudo-differential operators have made possible further progress in the study of elliptic boundary value problems and hence the study of parabolic initial boundary value problems. The presentation of these new results is the main purpose of this book.

We study a class of degenerate boundary value problems for secondorder elliptic differential operators in the framework of Lp Sobolev spaces which include as particular cases the Dirichlet and Neumann problems, and proves that these boundary value problems provide an interesting example of analytic semigroups in the Lp topology. As an application, we can apply these results to the initial boundary value problems for semilinear parabolic differential equations of second order in the framework of Lp spaces. We confined ourselves to the simple but important boundary condition. This makes it possible to develop our basic machinery with a minimum of bother and the principal ideas can be presented concretely and explicitly.

Let Ω be a bounded domain of Euclidean space Rn, with C∞ boundary Γ = ∂Ω; its closure is an n-dimensional, compact C∞ manifold with boundary.