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Author: Henri Berestycki
Publisher: Springer Verlag
ISBN: 9780387341583
Category : Mathematics
Languages : en
Pages : 410
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Book Description
The book is about reaction-diffusion equations in unbounded domains with a special emphasis on traveling waves and their generalizations as well as on different notions of propagation. It includes a general presentation of all the classical results in this area. Even for some well known results, in some cases, original proofs are included which are simpler and more elegant than the known ones. The book gives a fairly comprehensive and coherent account of the recent developments and current research in this active area. It also contains some of the basic results about elliptic and parabolic partial differential equations and a chapter on the different versions of the maximum principles. Thus, it also serves as an introduction to these topics. Each chapter is made as much autonomous as possible. Each one has a specific introduction as well as brief mentions of extensions or of related subjects. Some outstanding open problems are mentioned along the way. Each introduction states the goals of the chapter, some of its main results, the framework and indicates how the chapter is organized. The book is addressed to researchers and graduate students in mathematics, in particular in analysis, partial differential equations and applied mathematics. It will be of interest as well to researchers and graduate students concerned by mathematical modeling in physics and in biology. It is planed to be a reference book of lasting value with all the important results on a topic which is commonly used in these fields.
Author: Henri Berestycki
Publisher: Springer Verlag
ISBN: 9780387341583
Category : Mathematics
Languages : en
Pages : 410
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Book Description
The book is about reaction-diffusion equations in unbounded domains with a special emphasis on traveling waves and their generalizations as well as on different notions of propagation. It includes a general presentation of all the classical results in this area. Even for some well known results, in some cases, original proofs are included which are simpler and more elegant than the known ones. The book gives a fairly comprehensive and coherent account of the recent developments and current research in this active area. It also contains some of the basic results about elliptic and parabolic partial differential equations and a chapter on the different versions of the maximum principles. Thus, it also serves as an introduction to these topics. Each chapter is made as much autonomous as possible. Each one has a specific introduction as well as brief mentions of extensions or of related subjects. Some outstanding open problems are mentioned along the way. Each introduction states the goals of the chapter, some of its main results, the framework and indicates how the chapter is organized. The book is addressed to researchers and graduate students in mathematics, in particular in analysis, partial differential equations and applied mathematics. It will be of interest as well to researchers and graduate students concerned by mathematical modeling in physics and in biology. It is planed to be a reference book of lasting value with all the important results on a topic which is commonly used in these fields.
Author: John Rinzel
Publisher:
ISBN:
Category :
Languages : en
Pages : 63
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Book Description
Consideration is given to a system of reaction diffusion equations which have qualitative significance for several applications including nerve conduction and distributed chemical/biochemical systems. These equations are of the FitzHugh-Nagumo type and contain three parameters. For certain ranges of the parameters the system exhibits two stable singular points. A singular perturbation construction is given to illustrate that there may exist both pulse type and transition type traveling waves. A complete, rigorous, description of which of these waves exist for a given set of parameters and how the velocities of the waves vary with the parameters is given for the case when the system has a piecewise linear nonlinearity. Numerical results of solutions to these equations are also presented. These calculations illustrate how waves are generated from initial data, how they interact during collisions, and how they are influenced by local disturbances and boundary conditions.
Author: Weiwei Ding
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
This dissertation is concerned with propagation phenomena of some evolution equations in periodic habitats. The main results consist of the following two parts. In the first part, we investigate the spatial spreading phenomena of some spatially periodic integro-difference equations. Firstly, we establish a general theory on the existence of spreading speeds for noncompact evolution systems, under the hypothesis that the linearized systems have principal eigenvalues. Secondly, we introduce the notion of uniform irreducibility for finite Radon measures on the circle. It is shown that, any generalized convolution operator generated by such a measure admits a principal eigenvalue. Finally, applying the above general theories, we prove the existence of spreading speeds for some integro-difference equations with uniformly irreducible dispersal kernels. In the second part, we study the front propagation phenomena of spatially periodic reaction-diffusion equations with bistable nonlinearities. Firstly, we focus on the propagation solutions in the class of pulsating fronts. It is proved that, under various assumptions on the reaction terms, pulsating fronts exist when the spatial period is small or large. We also characterize the sign of the front speeds and we show the global exponential stability of the pulsating fronts with nonzero speed. Secondly, we investigate the propagation solutions in the larger class of transition fronts. It is shown that, under suitable assumptions, transition fronts are reduced to pulsating fronts with nonzero speed. But we also prove the existence of new types of transition fronts which are not pulsating fronts.
Author: Masaharu Taniguchi
Publisher:
ISBN: 9784864970976
Category : Mathematics
Languages : en
Pages : 0
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Book Description
The study on traveling fronts in reaction-diffusion equations is the first step to understand various kinds of propagation phenomena in reaction-diffusion models in natural science. One dimensional traveling fronts have been studied from the 1970s, and multidimensional ones have been studied from around 2005. This volume is a text book for graduate students to start their studies on traveling fronts. Using the phase plane analysis, we study the existence of traveling fronts in several kinds of reaction-diffusion equations. For a nonlinear reaction term, a bistable one is a typical one. For a bistable reaction-diffusion equation, we study the existence and stability of two-dimensional V-form fronts, and we also study pyramidal traveling fronts in three or higher space dimensions. The cross section of a pyramidal traveling front forms a convex polygon. It is known that the limit of a pyramidal traveling front gives a new multidimensional traveling front. For the study the multidimensional traveling front, studying properties of pyramidal traveling fronts plays an important role. In this volume, we study the existence, uniqueness and stability of a pyramidal traveling front as clearly as possible for further studies by graduate students. For a help of their studies, we briefly explain and prove the well-posedness of reaction-diffusion equations and the Schauder estimates and the maximum principles of solutions.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets
Author: Christopher Henderson
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
Reaction-advection-diffusion (RAD) equations are a class of non-linear parabolic equations which are used to model a diverse range of biological, physical, and chemical phenomena. Originally introduced in the early twentieth century as a model for population dynamics, they have been used in recent years in diverse contexts including climate change, criminal behavior, and combustion. These equations are characterized by the combination of three behaviors: spreading, stirring, and growth/decay. The main focus of mathematical research into RAD equations over the past century has been in characterizing the propagation of solutions. Indeed, these equations are characterized by the invasion of an unstable state by a stable state at a constant rate (for instance, the invasion of empty space by a population until the environmental carrying capacity is reached). In general, this can be characterized by the existence, uniqueness, and stability of traveling wave solutions, or solutions with a fixed profile which move at a constant speed in time. In general, the speed and shape of these traveling waves gives us the speed with which the stable state invades the unstable state. This thesis assumes the following trajectory, investigating two specific RAD equations: the Fisher-KPP equation, used in population dynamics, and a coupled reactive-Boussinesq system, used to model combustion in a fluid. For the former equation, we prove results regarding the precise spreading rate, and for the latter equation, we prove an existence result for a special solution that generalizes the traveling wave. In the first part of this thesis, we prove two results quantifying the precise speed of spreading for solutions to the Cauchy problem of the Fisher-KPP equation. The first of these results, concerning localized initial data, provides intuition for a lower order term obtained non-rigorously in. Specifically, we prove a quantitative convergence-to-equilibrium result in a related model, which has been used as a close approximation of the Fisher-KPP equation. The second of these results, concerning non-localized initial data and building on the work of Hamel and Roques, quantifies the super-linear in time spreading of the population. Here we compute the highest order term in the spreading for a broad class of initial data. In the second part of this thesis, we look at a coupled system that models combustion in a fluid, and we prove a qualitative propagation result. Unlike classical models, this relatively new system accounts for the effect of advection induced by the buoyancy force that results from the evolution of the temperature. Essentially, this means that we take into account the phenomenon that ``hot air rises.'' We exhibit a generalized traveling wave solution of this system, called a pulsating front, in two-dimensional periodic domains. To our knowledge, this is the first result regarding the existence of ``pulsating fronts'' in a coupled system.
Author: H Wilhelmsson
Publisher: CRC Press
ISBN: 1420033581
Category : Science
Languages : en
Pages : 170
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Book Description
The physics of hot plasmas is of great importance for describing many phenomena in the universe and is fundamental for the prospect of future fusion energy production on Earth. Nontrivial results of nonlinear electromagnetic effects in plasmas include the self-organization and self-formation in the plasma of structures compact in time and space. Th
Author: Andreas Liehr
Publisher: Springer Science & Business Media
ISBN: 3642312519
Category : Science
Languages : en
Pages : 227
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Book Description
Why writing a book about a specialized task of the large topic of complex systems? And who will read it? The answer is simple: The fascination for a didactically valuable point of view, the elegance of a closed concept and the lack of a comprehensive disquisition. The fascinating part is that field equations can have localized solutions exhibiting the typical characteristics of particles. Regarding the field equations this book focuses on, the field phenomenon of localized solutions can be described in the context of a particle formalism, which leads to a set of ordinary differential equations covering the time evolution of the position and the velocity of each particle. Moreover, starting from these particle dynamics and making the transition to many body systems, one considers typical phenomena of many body systems as shock waves and phase transitions, which themselves can be described as field phenomena. Such transitions between different level of modelling are well known from conservative systems, where localized solutions of quantum field theory lead to the mechanisms of elementary particle interaction and from this to field equations describing the properties of matter. However, in dissipative systems such transitions have not been considered yet, which is adjusted by the presented book. The elegance of a closed concept starts with the observation of self-organized current filaments in a semiconductor gas discharge system. These filaments move on random paths and exhibit certain particle features like scattering or the formation of bound states. Neither the reasons for the propagation of the filaments nor the laws of the interaction between the filaments can be registered by direct observations. Therefore a model is established, which is phenomenological in the first instance due to the complexity of the experimental system. This model allows to understand the existence of localized structures, their mechanisms of movement, and their interaction, at least, on a qualitative level. But this model is also the starting point for developing a data analysis method that enables the detection of movement and interaction mechanisms of the investigated localized solutions. The topic is rounded of by applying the data analysis to real experimental data and comparing the experimental observations to the predictions of the model. A comprehensive publication covering the interesting topic of localized solutions in reaction diffusion systems in its width and its relation to the well known phenomena of spirals and patterns does not yet exist, and this is the third reason for writing this book. Although the book focuses on a specific experimental system the model equations are as simple as possible so that the discussed methods should be adaptable to a large class of systems showing particle-like structures. Therefore, this book should attract not only the experienced scientist, who is interested in self-organization phenomena, but also the student, who would like to understand the investigation of a complex system on the basis of a continuous description.
Author: Antoine Pauthier
Publisher:
ISBN:
Category :
Languages : en
Pages : 131
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Book Description
The aim of this thesis is to study two examples of propagation phenomena in heterogeneous reaction-diffusion equations.The purpose of the first part is to understand the effect of nonlocal exchanges between a line of fast diffusion and a two dimensional environment in which reaction-diffusion of KPP type occurs. The initial model was introduced in 2013 by Berestycki, Roquejoffre, and Rossi. In the first chapter we investigate how the nonlocal coupling between the line and the plane enhances the spreading in the direction of the line; we also investigate how different exchange functions may maximize or not the spreading speed.The second chapter is concerned with the singular limit of nonlocal exchanges that tend to Dirac masses. We show the convergence of the dynamics in a rather strong sense. In the third chapter we study the limit of long range exchanges with constant mass. It gives an infimum for the asymptotic speed of spreading for these models that still could be bigger than the usual KPP spreading speed.The second part of this thesis is concerned with entire solutions for heterogeneous bistable equations.We consider a two dimensional domain infinite in one direction, bounded in the other, that converges to a cylinder as x goes to minus infinity. We prove the existence of an entire solution in such a domain which is the bistable wave for t tends to minus infinity. It also lead us to investigate a one dimensional model with a non-homogeneous reaction term,for which we prove the same property.
Author: Henri Berestycki
Publisher: Springer Science & Business Media
ISBN: 9781402009723
Category : Mathematics
Languages : en
Pages : 554
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Book Description
Nonlinear partial differential equations abound in modern physics. The problems arising in these fields lead to fascinating questions and, at the same time, progress in understanding the mathematical structures is of great importance to the models. Nevertheless, activity in one of the approaches is not always sufficiently in touch with developments in the other field. The book presents the joint efforts of mathematicians and physicists involved in modelling reactive flows, in particular superconductivity and superfluidity. Certain contributions are fundamental to an understanding of such cutting-edge research topics as rotating Bose-Einstein condensates, Kolmogorov-Zakharov solutions for weak turbulence equations, and the propagation of fronts in heterogeneous media.
Author: Brian H. Gilding
Publisher: Springer Science & Business Media
ISBN: 9783764370718
Category : Mathematics
Languages : en
Pages : 224
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Book Description
This monograph has grown out of research we started in 1987, although the foun dations were laid in the 1970's when both of us were working on our doctoral theses, trying to generalize the now classic paper of Oleinik, Kalashnikov and Chzhou on nonlinear degenerate diffusion. Brian worked under the guidance of Bert Peletier at the University of Sussex in Brighton, England, and, later at Delft University of Technology in the Netherlands on extending the earlier mathematics to include nonlinear convection; while Robert worked at Lomonosov State Univer sity in Moscow under the supervision of Anatolii Kalashnikov on generalizing the earlier mathematics to include nonlinear absorption. We first met at a conference held in Rome in 1985. In 1987 we met again in Madrid at the invitation of Ildefonso Diaz, where we were both staying at 'La Residencia'. As providence would have it, the University 'Complutense' closed down during this visit in response to student demonstra tions, and, we were very much left to our own devices. It was natural that we should gravitate to a research topic of common interest. This turned out to be the characterization of the phenomenon of finite speed of propagation for nonlin ear reaction-convection-diffusion equations. Brian had just completed some work on this topic for nonlinear diffusion-convection, while Robert had earlier done the same for nonlinear diffusion-absorption. There was no question but that we bundle our efforts on the general situation.
