This monograph provides new functional analytic developments on spectral problems in various applied fields such as jump equations, Kolmogorov differential equations, weighted graphs, neutron transport theory, population dynamics, linearized non-local Allen Cahn equations and perturbed convolution semigroups. With an emphasis on spectral problems (in Lebesgue spaces) connected to positivity, it uses powerful tools developed over the last two decades to unify and extend established results in various directions, offering new perspectives in a diverse range of subjects. Many open questions are scattered throughout, in the hope of promoting further research.
Covering topics lying at the intersection between functional analysis and partial differential equations, the book is primarily targeted toward applied mathematicians working in kinetic theory, probability theory, mathematical biology and, more generally, partial differential equations, who are interested in peripheral spectral theory. It may also draw pure mathematicians working in semigroup theory towards very rich applied areas. Although mainly aimed at professional mathematicians, it will also be useful to PhD students and post-doctoral researchers. The long chapter devoted to transport theory will be of interest to physicists or engineers involved in neutron transport.
Chapter 1. Introduction.- Chapter 2. Mathematical Toolbox.- Chapter 3. Time asymptotics of Kato-Voigt semigroups.- Chapter 4. Spectral analysis of Desch semigroups.- Chapter 5. Spectral analysis of absorption semigroups.- Chapter 6. On jump semigroups in L1(v).- Chapter 7. Detailed balance and Hilbert space theory.- Chapter 8. Perturbed Frobenius-Perron semigroups.- Chapter 9. Jump semigroups on countable state spaces.- Chapter 10. Transport Theory.- Chapter 11. Mathematical Population Dynamics.- Chapter 12. Linearized non local Allen-Cahn equations.- Chapter 13. Spectra of perturbed convolution semigroups.- Chapter 14. Miscellaneous.
Mustapha Mokhtar-Kharroubi earned his "Thèse d'État" (the former French Habilitation) from Pierre and Marie Curie University Paris VI in 1987. He has been a Professor at the University of Franche-Comté since 1990. His research interests include the spectral theory of kinetic equations, population dynamics, Schrödinger equations, and operator semigroups, among other topics. He has published over 80 articles in peer-reviewed mathematical journals, authored two monographs, and co-edited a volume based on a CIMPA School. He has supervised nine PhD theses, and seven of his former doctoral students currently hold positions as Associate Professors or Professors.
