Blow-up Theories for Semilinear Parabolic Equations
Blow-up Theories for Semilinear Parabolic Equations

Author: Bei Hu

Publisher: Springer Science & Business Media

Published: 2011-03-23

Total Pages: 137

ISBN-13: 3642184596

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There is an enormous amount of work in the literature about the blow-up behavior of evolution equations. It is our intention to introduce the theory by emphasizing the methods while seeking to avoid massive technical computations. To reach this goal, we use the simplest equation to illustrate the methods; these methods very often apply to more general equations.

Fractional-in-Time Semilinear Parabolic Equations and Applications
Fractional-in-Time Semilinear Parabolic Equations and Applications

Author: Ciprian G. Gal

Publisher: Springer Nature

Published: 2020-09-23

Total Pages: 193

ISBN-13: 3030450430

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This book provides a unified analysis and scheme for the existence and uniqueness of strong and mild solutions to certain fractional kinetic equations. This class of equations is characterized by the presence of a nonlinear time-dependent source, generally of arbitrary growth in the unknown function, a time derivative in the sense of Caputo and the presence of a large class of diffusion operators. The global regularity problem is then treated separately and the analysis is extended to some systems of fractional kinetic equations, including prey-predator models of Volterra–Lotka type and chemical reactions models, all of them possibly containing some fractional kinetics. Besides classical examples involving the Laplace operator, subject to standard (namely, Dirichlet, Neumann, Robin, dynamic/Wentzell and Steklov) boundary conditions, the framework also includes non-standard diffusion operators of "fractional" type, subject to appropriate boundary conditions. This book is aimed at graduate students and researchers in mathematics, physics, mathematical engineering and mathematical biology, whose research involves partial differential equations.

Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics
Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics

Author: Tian Ma

Publisher: American Mathematical Soc.

Published: 2005

Total Pages: 248

ISBN-13: 0821836935

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This monograph presents a geometric theory for incompressible flow and its applications to fluid dynamics. The main objective is to study the stability and transitions of the structure of incompressible flows and its applications to fluid dynamics and geophysical fluid dynamics. The development of the theory and its applications goes well beyond its original motivation of the study of oceanic dynamics. The authors present a substantial advance in the use of geometric and topological methods to analyze and classify incompressible fluid flows. The approach introduces genuinely innovative ideas to the study of the partial differential equations of fluid dynamics. One particularly useful development is a rigorous theory for boundary layer separation of incompressible fluids. The study of incompressible flows has two major interconnected parts. The first is the development of a global geometric theory of divergence-free fields on general two-dimensional compact manifolds. The second is the study of the structure of velocity fields for two-dimensional incompressible fluid flows governed by the Navier-Stokes equations or the Euler equations. Motivated by the study of problems in geophysical fluid dynamics, the program of research in this book seeks to develop a new mathematical theory, maintaining close links to physics along the way. In return, the theory is applied to physical problems, with more problems yet to be explored. The material is suitable for researchers and advanced graduate students interested in nonlinear PDEs and fluid dynamics.

Global Solution Curves For Semilinear Elliptic Equations
Global Solution Curves For Semilinear Elliptic Equations

Author: Philip Korman

Publisher: World Scientific

Published: 2012-02-10

Total Pages: 254

ISBN-13: 9814458066

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This book provides an introduction to the bifurcation theory approach to global solution curves and studies the exact multiplicity of solutions for semilinear Dirichlet problems, aiming to obtain a complete understanding of the solution set. This understanding opens the way to efficient computation of all solutions. Detailed results are obtained in case of circular domains, and some results for general domains are also presented.The author is one of the original contributors to the field of exact multiplicity results.

Optimal Control of Differential Equations
Optimal Control of Differential Equations

Author: Nicolae H. Pavel

Publisher: CRC Press

Published: 2020-08-19

Total Pages: 356

ISBN-13: 1000153770

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"Based on the International Conference on Optimal Control of Differential Equations held recently at Ohio University, Athens, this Festschrift to honor the sixty-fifth birthday of Constantin Corduneanu an outstanding researcher in differential and integral equations provides in-depth coverage of recent advances, applications, and open problems relevant to mathematics and physics. Introduces new results as well as novel methods and techniques!"

Dynamics of Evolutionary Equations
Dynamics of Evolutionary Equations

Author: George R. Sell

Publisher: Springer Science & Business Media

Published: 2002-01-02

Total Pages: 692

ISBN-13: 9780387983479

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The theory and applications of infinite dimensional dynamical systems have attracted the attention of scientists for quite some time. Dynamical issues arise in equations that attempt to model phenomena that change with time. The infi nite dimensional aspects occur when forces that describe the motion depend on spatial variables, or on the history of the motion. In the case of spatially depen dent problems, the model equations are generally partial differential equations, and problems that depend on the past give rise to differential-delay equations. Because the nonlinearities occurring in thse equations need not be small, one needs good dynamical theories to understand the longtime behavior of solutions. Our basic objective in writing this book is to prepare an entree for scholars who are beginning their journey into the world of dynamical systems, especially in infinite dimensional spaces. In order to accomplish this, we start with the key concepts of a semiflow and a flow. As is well known, the basic elements of dynamical systems, such as the theory of attractors and other invariant sets, have their origins here.