JUCS - Journal of Universal Computer Science 15(6): 1301-1316, doi: 10.3217/jucs-015-06-1301

Topological Complexity of Blowup Problems

‡ FernUniversität Hagen, Hagen, Germany§ University of Cincinnati, Cincinnati, United States of America

Corresponding author: Robert Rettinger ( robert.rettinger@fernuni-hagen.de ) © Robert Rettinger, Klaus Weihrauch, Ning Zhong. This article is freely available under the J.UCS Open Content License. Citation:
Rettinger R, Weihrauch K, Zhong N (2009) Topological Complexity of Blowup Problems. JUCS - Journal of Universal Computer Science 15(6): 1301-1316. https://doi.org/10.3217/jucs-015-06-1301 |

Abstract

Consider the initial value problem of the first-order ordinary differential equation where the locally Lipschitz continuous function with open domain and the initial datum (t0, x0) ∈ are given. It is shown that the solution operator producing the maximal "time" interval of existence and the solution on it is computable. Furthermore, the topological complexity of the blowup problem is studied for functions f defined on the whole space. For each such function f the set Z of initial conditions (t0, x0) for which the positive solution does not blow up in finite time is a -set. There is even a computable operator determining Z from f. For l ≥ 2 this upper -complexity bound is sharp. For l = 1 the blowup problem is simpler.

Keywords

Type-2 theory, differential equation, blowup