Abstract
The equationx (n)(t)=(−1)n│x(t)│k withk>1 is considered. In the casen≦4 it is proved that solutions defined in a neighbourhood of infinity coincide withC(t−t0)−n/(k−1), whereC is a constant depending only onn andk. In the general case such solutions are Kneser solutions and can be estimated from above and below by a constant times (t−t 0)−n/(k−1). It is shown that they do not necessarily coincide withC(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.
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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.
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Kozlov, V.A. On Kneser solutions of higher order nonlinear ordinary differential equations. Ark. Mat. 37, 305–322 (1999). https://doi.org/10.1007/BF02412217
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DOI: https://doi.org/10.1007/BF02412217