Theory And Applications Of Abstract Semilinear Cauchy Problems
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Theory and Applications of Abstract Semilinear Cauchy Problems
Several types of differential equations, such as functional differential equation, age-structured models, transport equations, reaction-diffusion equations, and partial differential equations with delay, can be formulated as abstract Cauchy problems with non-dense domain. This monograph provides a self-contained and comprehensive presentation of the fundamental theory of non-densely defined semilinear Cauchy problems and their applications. Starting from the classical Hille-Yosida theorem, semigroup method, and spectral theory, this monograph introduces the abstract Cauchy problems with non-dense domain, integrated semigroups, the existence of integrated solutions, positivity of solutions, Lipschitz perturbation, differentiability of solutions with respect to the state variable, and time differentiability of solutions. Combining the functional analysis method and bifurcation approach in dynamical systems, then the nonlinear dynamics such as the stability of equilibria, center manifold theory, Hopf bifurcation, and normal form theory are established for abstract Cauchy problems with non-dense domain. Finally applications to functional differential equations, age-structured models, and parabolic equations are presented. This monograph will be very valuable for graduate students and researchers in the fields of abstract Cauchy problems, infinite dimensional dynamical systems, and their applications in biological, chemical, medical, and physical problems.
Problems in Mathematical Biophysics
The book "Problems in Mathematical Biophysics - a volume in memory of Alberto Gandolfi" aims at reviewing the current state of the art of the mathematical approach to various areas of theoretical biophysics. Leading authors in the field have been invited to contribute, having a strong appreciation of Alberto Gandolfi as a scientist and as a man and sharing his same passion for biology and medicine, as well as his style of investigation. Encompassing both theoretical and practical aspects of Mathematical Biophysics, the topics covered in this book span a spectrum of different problems, in biology, and medicine, including population dynamics, tumor growth and control, immunology, epidemiology, ecology, and others. As a result, the book offers a comprehensive and current overview of compelling subjects and challenges within the realm of mathematical biophysics. In their contributions, the authors have effectively conveyed not only their research findings but also their peculiar perspective and approach to problem-solving, dealing with oncology, epidemiology, neuro-sciences, and biochemistry. The chapters pertain to a wide array of mathematical areas such as continuous Markov chains, partial differential equations, kinetic theory, applied statistical mechanics, noise-induced transitions, and many others.
Almost Periodicity and Almost Automorphy
Author: Abdallah Afoukal
language: en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date: 2025-03-03
When we study differential equations in Banach spaces whose coefficients are linear unbounded operators, we feel that we are working in ordinary differential equations; however, the fact that the operator coefficients are unbounded makes things quite different from what is known in the classical case. Examples or applications for such equations are naturally found in the theory of partial differential equations. More specifically, if we give importance to the time variable at the expense of the spatial variables, we obtain an “ordinary differential equation” with respect to the variable which was put in evidence. Thus, for example, the heat or the wave equation gives rise to ordinary differential equations of this kind. Adding boundary conditions can often be translated in terms of considering solutions in some convenient functional Banach space. The theory of semigroups of operators provides an elegant approach to study this kind of systems. Therefore, we can frequently guess or even prove theorems on differential equations in Banach spaces looking at a corresponding pattern in finite dimensional ordinary differential equations.