A careful and accessible exposition of a functional analytic approach to initial boundary value problems for semilinear parabolic differential equations, with a focus on the relationship between analytic semigroups and initial boundary value problems. This semigroup approach is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of pseudo-differential operators, one of the most influential works in the modern history of analysis. Complete with ample illustrations and additional references, this new edition offers both streamlined analysis and better coverage of important examples and applications. A powerful method for the study of elliptic boundary value problems, capable of further extensive development, is provided for advanced undergraduates or beginning graduate students, as well as mathematicians with an interest in functional analysis and partial differential equations.
Kazuaki Taira is a part-time professor at Waseda University, Japan. He previously worked as a professor of mathematics at Hiroshima University from 1995 until 1998 and at the University of Tsukuba from 1998 until 2009. Taira has authored many books, the most recent being the second edition of Semigroups, Boundary Value Problems and Markov Processes (2014).
1. Introduction and main results; 2. Preliminaries from functional analysis; 3. Theory of analytic semigroups; 4. Sobolev imbedding theorems; 5. Lp theory of pseudo-differential operators; 6. Lp approach to elliptic boundary value problems; 7. Proof of theorem 1.1; 8. Proof of theorem 1.2; 9. Proof of theorems 1.3 and 1.4; Appendix A. The Laplace Transform; Appendix B. The Maximum Principle; Appendix C. Vector bundles; References; Index.