Global Solution Branches of Two Point Boundary Value Problems
The book deals with parameter dependent problems of the form u"+*f(u)=0 on an interval with homogeneous Dirichlet or Neuman boundary conditions. These problems have a family of solution curves in the (u,*)-space. By examining the so-called time maps of th
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1458
Renate Schaaf
Global Solution Branches of Two Point Boundary Value Problems
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona
Lecture Notes in Mathematics Edited by A. Dold, B. Eckmann and F. Takens
1458
Renate Schaaf
Global Solution Branches of Two Point Boundary Value Problems
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona
Author
Renate Schaaf Utah State University, Department of Mathematics and Statistics Logan, UT 84322-3900, USA
Mathematics Subject Classification (1980): 34B15, 35B32, 34C25 ISBN 3-540-53514-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-53514-4 Springer-Verlag New York Berlin Heidelberg
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Introduction For the parameter dependent problem
u"(x) + >.2 f(u(x)) = 0
(I-I-I)
U(O) = u(l) = 0
the following is known from application of general local ([13]) or global ([24]) bifurcation theorems in the case f(O) = 0, 1'(0) > 0 : The trivial solution u == 0 exists for each >.. From this trivial solution branch there is bifurcation of nontrivial solutions in each point (0, >.) for which the linearized problem (I-I-I) has a nontrivial kernel, i.e., for>' = i1rfJp(O) , i = 1,2, .... The bifurcating branches, i.e., connected components of nontrivial solutions with bifurcation points in the (>., u) -space, are all unbounded and each branch consists of solutions (>., u) where u has a number of simple zeroes in ] 0, 1 [ which is characteristic for the branch. In the solution branches bifurcating from 0 all u are bounded by the first positive and first negative zero of' f . In applications more information about the shape of solution branches is needed. It is easy to see that all branches are in fact curves which are at least as smooth as f is (see below). It is then of interest whether these curves have turns with respect to the >. -direction or not, and if so, how many turning points there are and where they are located. Let, e.g., the branch of positive and negative solutions to (I-I-I) look like this:
u
Figure 1.1.1
,IV
Introduction
Then (I-I-I) is the stationary equation of
(1-1-2)
Ut
=
Uxx
+..\2 feu)
u(t,O) = u(t, 1) = 0
and the directions of the solution branches determine the stability of the stationary states as indicated. In combustion problems there often occurs an equation of t
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