Nonlinear Diffusion Problems

262 Pages·1976·42.456 MB·English

by  diekmann, temme

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MC SYLLABUS 28
0. DIEKMANN
N.M. TEMME (EDS)
NONLINEAR
DIFFUSION PROBLEMS
SECOND PRINTING
MATHEMATISCH CENTRUM AMSTER DAM 1982
AMS(MOS) Subject classification scheme (1970) 35-02,35BXX,35K55,34C35,92A05,92A10
ISBN 90 6196 126 2
First printing: 1976
Second printing: 1982
v
CONTENTS
Introduction vii
I. MODELS AND METHODS: A FIRST IMPRESSION
II. EXISTENCE AND UNIQUENESS OF ONE-DIMENSIONAL
NONLINEAR PARABOLIC EQUATIONS 26
III. MONOTONE ITERATION 61
IV. TRAVELLING FRONTS 84
V. EXAMINATION OF STABILITY BY MEANS OF LYAPUNOV FUNCTIONS 136
VI. MATHEMATICAL ASPECTS OF A MODEL CHEMICAL
REACTION WITH DIFFUSION 165
VII. NERVE AXON EQUATIONS 233
REFERENCES 243
GENERAL INDEX 245

vii
INTRODUCTION
In the first half of 1976, the department of Applied Mathematics of
the Mathematical Centre organized a colloquium on Nonlinear Diffusion
Problems under supervision of H. A. Lauwerier and L.A. Peletier. This book
contains the proceedings of the lectures of that colloquium.
At the moment many papers about nonlinear diffusion problems are
appearing in the literature. The idea of the colloquium was to present back
ground material on the qualitative analysis of parabolic partial differen
tial equations in such a way that the existing and forthcoming literature
was made accessible to the audience. Theoretical methods were presented and,
whenever possible, their use was demonstrated in the analysis of some pro
totype problem. Most of these prototype problems arise as mathematical mo
dels of a biological phenomenon and the mathematical questions are motivated
by these models. Thi.s is a reflection of the fact that quite often the re
search in this area is inspired by biological applications.
In Chapter I, first a brief indication of the biological background
of some mathematical questions is presented and subsequently the use of the
variational method of proving existence of equilibria and the Lyapunov
method of proving stability are demonstrated.
Chapter II contains basic results on existence and uniqueness of solu
tions of the initial-boundary value problem for a one-dimensional nonlinear
parabolic equation. Moreover the maximum principle is introduced.
The maximum principle is used extensively in Chapter III. It is shown
how upper and lower solutions yield monotone iteration schemes and how in
this way one can get information concerning existence, nonexistence and
stability of equilibria.
Chapter IV is devoted to travelling wave solutions (i.e., solutions of
the form u(x,t) w(x-ct) of the one-dimensional equation ut = uxx + f(u),
-00 < x < 00• Phase plane methods are used to show existence and the maximum
principle and Lyapunov functions to obtain results concerning stability.
In Chapter V the basic theory of Lyapunov functions for dynamical sys
tems is presented and subsequently it is shown how this theory can be
applied to parabolic initial-boundary value problems (again in the case of
one space-variable) .
viii
The last two chapters deal with systems of nonlinear diffusion equa
tions. In Chapter VI the system resulting from a model chemical reaction,
as proposed, by Prigogine is discussed. The existence for all time of a
solution of the initial-boundary value problem is proved by means of the
Faedo-Galerkin method. Then, following the general lines of bifurcation
theory, the existence and stability of equilibria (and of periodic solu
tions) is discussed by analyzing their dependence on a parameter.
Finally, in Chapter VII an indication is given of the questions, re
sults and techniques occurring in the rapidly increasing mathematical lite
rature inspired by the Hodgkin-Huxley theory of the propagation of an elec
trical impulse along a nerve axon.
The results in this book are not new. Apart from minor details all of
the contents can be found in the literature. However, it is hoped that for
everybody who is interested in the subject, this book can serve as a use
ful and quick introduction.
The lectures were prepared in working groups and every member of our
department has contributed to the success of the colloquium.
The lectures were given by
o. DIEKMANN Chapter I
J.W. de ROEVER Chapter II
I.G. SPRINKHUIZEN-KUYPER Chapter III
B. DIJKHUIS Chapter IV 1-3
E.J.M. VELING Chapter IV 4-5
T.H. KOORNWINDER Chapter v
T.M.T. COO LEN Chapter VI 1-9
J. GRASMAN Chapter VI 10-15
E.J.M. VELING Chapter VII
They also are the authors of the corresponding parts of the text. For
Chapter IV 1-3, O. Diekmann and E.J.M. Veling were co-authors. The pictures
were made by G.J.M. Laan. In the selection of the material the guidance of
L.A. Peletier was very helpful.
I. MODELS AND METHODS: A FIRST IMPRESSION
1. INTRODUCTION
A frequent approach in applied mathematics is to study prototype prob
lems in full detail. In doing so, one hopes to get examples of both the qual
itative behaviour of solutions of a whole class of problems, and the use of
analytical tools in a specific case.
For linear partial differential equations there is the classification
into equations of elliptic, parabolic and hyperbolic type, and each type has
its classical representative (the equation of Laplace, the heat equation and
the wave equation). The study of these equations has resulted in a clear view
of the behaviour of their solutions, and this in turn has led to an under
standing of the physical processes that are modelled by the equations.
The diversity of phenomena that one can expect in the field of nonlinear
partial differential equations is much larger (for example, the important
property of superposition of solutions is lost). Accordingly, there will be
a lot of prototype problems, but unfortunately one does not always know in
advance which problems deserve this qualification.
In this series of lectures a number of nonlinear diffusion problems will
be discussed which have had much attention in the recent mathematical litera
ture, and which are likely to remain in the center of interest for some time
to come. Some of the problems are mathematical models of phenomena from the
natural sciences (in fact biology), and some are reduced forms of such mod
els. The objective is to analyze equations which are as simple as possible
and yet show the most important qualitative features.
It is hoped that the meaning of "nonlinear diffusion problem" is intu
itively clear and we will not try to define it (we would certainly get en
tangled). Moreover, we emphasize that there are a lot of nonlinear diffusion
problems which are rather different from the ones that will be dealt with
here (for examples we refer to [1]).
In the next three sections of this first chapter we will give a brief
indication of the biological background of the mathematical questions. In
the remaining sections one of the problems is worked out and the use of
general methods is demonstrated.
2
2. CONDUCTION OF A NERVE IMPULSE
How is information transmitted through the nervous system? Experimental
ly it is observed that electrical impulses travel along individual nerve
cells. These impulses have the form shown in Figure 1 and they are propagated
with constant velocity.
Figure 1
Hodgkin and Huxley developed a theoretical model which can be formulated
mathematically as:
uxx ut + I(u,w),
(2.1) -oo < x < oo, 0 < t < oo,
wt P(u)w + q(u),
where subscripts denote partial differentiation. Here u(x,t) denotes the
electrical potential across the membrane that surrounds the (infinitely
long, cylindrical) nerve cell and w(x,t) is a three-dimensional vector which
describes the permeability of the membrane to certain ions. I, P and q are
given functions (some of the functional dependence on u and w is more or
less assumed ad hoe, since the molecular structure of the membrane is not
completely known) .
The equations (2.1) are very complicated and there is not much hope of
a useful analytical treatment. The results of extensive numerical studies
are in good agreement with the experiments, but one way or another computer
simulation on its own is not fully satisfactory. The phenomenon of a so
called travelling wave solution (i.e., a solution depending only on the
single variable s = x + et, where c F 0 is constant) is not observed for
linear parabolic partial differential equations (although of course the un-