Arkiv för Matematik
- Ark. Mat.
- Volume 37, Number 2 (1999), 305-322.
On Kneser solutions of higher order nonlinear ordinary differential equations
The equation x(n)(t)=(−1)n│x(t)│k with k>1 is considered. In the case n≦4 it is proved that solutions defined in a neighbourhood of infinity coincide with C(t−t0)−n/(k−1), where C is a constant depending only on n and k. In the general case such solutions are Kneser solutions and can be estimated from above and below by a constant times (t−t0)−n/(k−1). It is shown that they do not necessarily coincide with C(t−t0)−n/(k−1). This gives a negative answer to two conjectures posed by Kiguradze that Kneser solutions are determined by their value in a point and that blow-up solutions have prescribed asymptotics.
Dedicated to Professor Vladimir Maz'ya on the occasion of his 60th birthday.
The author was supported by the Swedish Natural Science Research Council (NFR) grant M-AA/MA 10879-304.
Ark. Mat., Volume 37, Number 2 (1999), 305-322.
Received: 9 October 1997
Revised: 20 October 1998
First available in Project Euclid: 31 January 2017
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1999 © Institut Mittag-Leffler
Kozlov, Vladimir A. On Kneser solutions of higher order nonlinear ordinary differential equations. Ark. Mat. 37 (1999), no. 2, 305--322. doi:10.1007/BF02412217. https://projecteuclid.org/euclid.afm/1485898637