Local Existence For The Cauchy Problem Of A Reaction Diffusion System With Discontinuous Nonlinearity
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Local Existence For The Cauchy Problem Of A Reaction Diffusion System With Discontinuous Nonlinearity
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Author : David Terman
language : en
Publisher:
Release Date : 1981
Local Existence For The Cauchy Problem Of A Reaction Diffusion System With Discontinuous Nonlinearity written by David Terman and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1981 with categories.
The most famous model for nerve conduction is due to Hodgkin and Huxley. However, a mathematical analysis of their model has proven very difficult. The complexity of the Hodgkin and Huxley model has led a number of other authors to introduce simpler models. In this report we consider one such simplification. It has been demonstrated (experimentally) that impulses in the nerve axon travel with constant shape and velocity. Mathematically, this corresponds to traveling wave solutions. A number of authors have proven that the mathematical equations considered here do possess traveling wave solutions. Another property of impulses in the nerve axon is the existence of a threshold phenomenon. This corresponds to the biological fact that a minimum stimulus is needed to trigger an impulse. Here we prove some preliminary results which will be used in a later report when it is demonstrated that the equations under study do indeed exhibit a threshold phenomenon.
Scientific And Technical Aerospace Reports
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Author :
language : en
Publisher:
Release Date : 1995
Scientific And Technical Aerospace Reports written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995 with Aeronautics categories.
Degenerate Diffusions
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Author : Panagiota Daskalopoulos
language : en
Publisher: European Mathematical Society
Release Date : 2007
Degenerate Diffusions written by Panagiota Daskalopoulos and has been published by European Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007 with Mathematics categories.
The book deals with the existence, uniqueness, regularity, and asymptotic behavior of solutions to the initial value problem (Cauchy problem) and the initial-Dirichlet problem for a class of degenerate diffusions modeled on the porous medium type equation $u_t = \Delta u^m$, $m \geq 0$, $u \geq 0$. Such models arise in plasma physics, diffusion through porous media, thin liquid film dynamics, as well as in geometric flows such as the Ricci flow on surfaces and the Yamabe flow. The approach presented to these problems uses local regularity estimates and Harnack type inequalities, which yield compactness for families of solutions. The theory is quite complete in the slow diffusion case ($ m>1$) and in the supercritical fast diffusion case ($m_c
On Cauchy Problems Of Reaction Diffusion Equations
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Author : Xuefeng Wang
language : en
Publisher:
Release Date : 1990
On Cauchy Problems Of Reaction Diffusion Equations written by Xuefeng Wang and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1990 with categories.
Energy Methods For Reaction Diffusion Problems
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Author : Xing Zhong
language : en
Publisher:
Release Date : 2012
Energy Methods For Reaction Diffusion Problems written by Xing Zhong and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with categories.
Nonlinear reaction-diffusion equations arise in many areas of applied sciences such as combustion modeling, population dynamics, chemical kinetics, etc. A fundamental problem in the studies of these equations is to understand the long time behavior of solutions of the associated Cauchy problem. These kinds of questions were originally studied in the context of combustion modeling. For suitable nonlinearity and a monotone increasing one-parameter family of initial data starting with zero data, small values of the parameter lead to extinction, whereas large values of the parameter may lead to spreading, i.e., the solution converging locally uniformly to a positive spatially independent stable steady state. A natural question is the existence of the threshold set of the parameters for which neither extinction nor spreading occurs. Even in one space dimension, this long standing question concerning threshold phenomena is far from trivial. Recent results show that if the initial data are compactly supported, then there exists a sharp transition between extinction and spreading, i.e., the threshold set contains only one point. However, these results rely in an essential way on compactly supported initial data assumption and only give limited information about the solutions when spreading occurs. In this dissertation, energy methods based on the gradient flow structure of reaction-diffusion equations are developed. The long time behavior of solutions of the Cauchy problem for nonlinear reaction-diffusion equations in one space dimension with the nonlinearity of bistable, ignition or monostable type is analyzed. For symmetric decreasing initial data in L2 (R) n L8 (R), the convergence results for the considered equations are studied, and the existence of a one-to-one relation between the long time behavior of the solution and the limit value of its energy is proved. In addition, by employing a weighted energy functional, a mathematical description of the equivalence between spreading and propagation of the solutions of the considered equations is given. More precisely, if spreading occurs, then the leading and the trailing edge of the solution propagate faster than some constant speed. Therefore, if the solution spreads, it also propagates. Furthermore, for a monotone family of symmetric decreasing initial data, there exists a sharp threshold between propagation and extinction.
Mrc Technical Summary Report
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Author : United States. Army. Mathematics Research Center
language : en
Publisher:
Release Date : 1984
Mrc Technical Summary Report written by United States. Army. Mathematics Research Center and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1984 with Mathematics categories.
Mrc Technical Summary Report
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Author : University of Wisconsin--Madison. Mathematics Research Center
language : en
Publisher:
Release Date : 1981
Mrc Technical Summary Report written by University of Wisconsin--Madison. Mathematics Research Center and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1981 with Applied mathematics categories.
Degenerate Nonlinear Diffusion Equations
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Author : Angelo Favini
language : en
Publisher: Springer
Release Date : 2012-05-08
Degenerate Nonlinear Diffusion Equations written by Angelo Favini and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-05-08 with Mathematics categories.
The aim of these notes is to include in a uniform presentation style several topics related to the theory of degenerate nonlinear diffusion equations, treated in the mathematical framework of evolution equations with multivalued m-accretive operators in Hilbert spaces. The problems concern nonlinear parabolic equations involving two cases of degeneracy. More precisely, one case is due to the vanishing of the time derivative coefficient and the other is provided by the vanishing of the diffusion coefficient on subsets of positive measure of the domain. From the mathematical point of view the results presented in these notes can be considered as general results in the theory of degenerate nonlinear diffusion equations. However, this work does not seek to present an exhaustive study of degenerate diffusion equations, but rather to emphasize some rigorous and efficient techniques for approaching various problems involving degenerate nonlinear diffusion equations, such as well-posedness, periodic solutions, asymptotic behaviour, discretization schemes, coefficient identification, and to introduce relevant solving methods for each of them.
Publications Of The Mathematics Research Center July 1 1975 Thru December 31 1985
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Author : University of Wisconsin--Madison. Mathematics Research Center
language : en
Publisher:
Release Date : 1985
Publications Of The Mathematics Research Center July 1 1975 Thru December 31 1985 written by University of Wisconsin--Madison. Mathematics Research Center and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1985 with Mathematics categories.
Technical Abstract Bulletin
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Author :
language : en
Publisher:
Release Date : 1981
Technical Abstract Bulletin written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1981 with Science categories.