Explore the fascinating intersection of mathematics and combustion theory in this comprehensive monograph, inspired by the pioneering work of N. N. Semenov and D. A. Frank-Kamenetskii. Delving into the nonlinear functional analytic approach, this book examines semilinear elliptic boundary value problems governed by the Arrhenius equation and Newton's law of heat exchange.

Key topics include:

• Detailed analysis of boundary conditions, including isothermal (Dirichlet) and adiabatic (Neumann) cases.
• Critical insights into ignition and extinction phenomena in stable steady temperature profiles, linked to the Frank-Kamenetskii parameter.

• Sufficient conditions for multiple positive solutions, revealing the S-shaped bifurcation curves of these problems.
  Designed for researchers and advanced students, this monograph provides a deep understanding of nonlinear functional analysis and elliptic boundary value problems through their application to combustion and chemical reactor models. Featuring detailed illustrations, clearly labeled figures, and tables, this book ensures clarity and enhances comprehension of complex concepts.

  Whether you are exploring combustion theory, functional analysis, or applied mathematics, this text offers profound insights and a thorough mathematical foundation.

  Provides a mathematical theory of combustion, which is inspired by N. N. Semenov and D. A. Frank-Kamenetskii Focuses on a relationship between chemical reactions and semilinear elliptic problems via nonlinear functional analysis Includes not only interesting mathematical results, but also interpretations of them from combustion theory

  Autorentext

  Kazuaki TAIRA is Professor of Mathematics at the University of Tsukuba, Japan, where he has taught since 1998. He received his Bachelor of Science (1969) degree from the University of Tokyo, Japan, and his Master of Science (1972) degree from Tokyo Institute of Technology, Japan, where he served as an Assistant between 1972-1978. He holds the Doctor of Science (1976) degree from the University of Tokyo, and the Doctorat d'Etat (1978) degree from Université de Paris-Sud, France, where he received a French Government Scholarship in 1976-1978. Dr. Taira was also a member of the Institute for Advanced Study, U. S. A., in 1980-1981. He was Associate Professor of the University of Tsukuba between 1981-1995, and Professor of Hiroshima University, Japan, between 1995-1998. His current research interests are in the study of three interrelated subjects in analysis: semigroups, elliptic boundary value problems and Markov processes.

  Inhalt

Preface.- Introduction and Main Results.- Part I. A Short Course in Nonlinear Functional Analysis.- Elements of Degree Theory.- Theory of Positive Mappings in Ordered Banach Spaces.- Elements of Bifurcation Theory.- Part II. Introduction to Semilinear Elliptic Problems via Semenov Approximation.- Elements of Functions Spaces.- Semilinear Hypoelliptic Robin Problems via Semenov Approximation.- Spectral Analysis of the Closed Realization A.- Local Bifurcation Theorem for Problem (6.4).- Fixed Point Theorems in Ordered Banach Spaces.- The Super-subsolution Method.- Sublinear Hypoelliptic Robin Problems.- Part III. A Combustion Problem with General Arrhenius Equations and Newtonian Cooling.- Proof of Theorem 1.5 (Existence and Uniqueness).- Proof of Theorem 1.7 (Multiplicity).- Proof of Theorem 1.9 (Unique solvability for sufficiently small).- Proof of Theorem 1.10 (Unique solvability for sufficiently large).- Proof of Theorem 1.11 (Asymptotics).- Part IV. Summary and Discussion.- Open Problems in Numerical Analysis.- Concluding Remarks.- Part V Appendix.- A The Maximum Principle for Second Order Elliptic Operators.- Bibliography.- Index.