Singular Initial Value Problems for Quasi-Linear Ordinary
Differential Equations
|
Co-author(s): Matthias Seiß |
Reference: Journal of Differential Equations, 281 (2021) 258-288 |
Description: We study initial value problems of scalar quasi-linear ordinary
differential equations where the initial point is an impasse point of the
equation. We first recall some notions from the classical geometric
singularity theory of differential equations and then show that
quasi-linear equations need their own theory leading to the notion of an
impasse point. In particular, we demonstrate that there might be impasse
points even if no singularities in the classical sense are present. We reduce
questions of existence, (non-)uniqueness and regularity of solutions of
initial value problems at impasse points to the problem of analysing the locals
solution of a vector field around a stationary point, i.e. to a classical
problem in dynamical system theory. In the second part of the paper, we use
the developed theory for a detailed analysis of the initial value problem of
the second-order equation g(x)u''=f(x,u,u') with the initial conditions
prescribed at a simple zero of g and recover geometrically the results
obtained by Liang with classical analytical techniques. |
PDF File: (806 kB)
|
Home,
Last update:
Thu May 27 20:26:48 2021
|