Abstract and Applied Analysis
Volume 2 (1997), Issue 3-4, Pages 239-256
Weakly hyperbolic equations with time degeneracy in Sobolev spaces
Faculty for Mathematics and Computer Sciences, Technical University Bergakademie Freiberg, Bernhard von Cotta Str.2, Freiberg 09596, Germany
Received 26 September 1997
Copyright © 1997 Michael Reissig. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The theory of nonlinear weakly hyperbolic equations was developed during the last decade in an astonishing way. Today we have a good
overview about assumptions which guarantee local well posedness in
spaces of smooth functions , Gevrey). But the situation is completely unclear in the case of Sobolev spaces. Examples from the linear theory show that in opposite to the strictly hyperbolic case we have in general no solutions valued in Sobolev spaces. If so-called Levi conditions are satisfied, then the situation will
be better. Using sharp Levi conditions of -type leads to an interesting effect. The linear weakly hyperbolic Cauchy problem has a Sobolev solution if the data are sufficiently smooth. The loss of derivatives will be determined in essential by special lower order terms. In the present paper we show that it is even possible to prove the existence of Sobolev solutions in the quasilinear case although one has the finite loss of derivatives for the linear case. Some of the tools are a reduction process to problems with special asymptotical behaviour, a Gronwall type lemma for differential inequalities with a singular coefficient, energy estimates and a fixed point argument.