Weakly hyperbolic equations with time degeneracy in Sobolev spaces
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 (C∞, Gevrey). But the situation is completely unclear in the case of Sobo...
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| Format: | Article |
| Language: | English |
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Wiley
1997-01-01
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| Series: | Abstract and Applied Analysis |
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| Online Access: | http://dx.doi.org/10.1155/S1085337597000377 |
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| _version_ | 1849468003677110272 |
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| author | Michael Reissig |
| author_facet | Michael Reissig |
| author_sort | Michael Reissig |
| collection | DOAJ |
| description | 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 (C∞, 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 C∞-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. |
| format | Article |
| id | doaj-art-df629f9b5c8b4784b9316f742fc55cd4 |
| institution | Kabale University |
| issn | 1085-3375 |
| language | English |
| publishDate | 1997-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-df629f9b5c8b4784b9316f742fc55cd42025-08-20T03:25:59ZengWileyAbstract and Applied Analysis1085-33751997-01-0123-423925610.1155/S1085337597000377Weakly hyperbolic equations with time degeneracy in Sobolev spacesMichael Reissig0Faculty for Mathematics and Computer Sciences, Technical University Bergakademie Freiberg, Bernhard von Cotta Str.2, Freiberg 09596, GermanyThe 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 (C∞, 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 C∞-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.http://dx.doi.org/10.1155/S1085337597000377Quasilinear weakly hyperbolic equationstime degeneracylocal existenceLevi conditionsSobolev spacesenergy method. |
| spellingShingle | Michael Reissig Weakly hyperbolic equations with time degeneracy in Sobolev spaces Abstract and Applied Analysis Quasilinear weakly hyperbolic equations time degeneracy local existence Levi conditions Sobolev spaces energy method. |
| title | Weakly hyperbolic equations with time degeneracy in Sobolev spaces |
| title_full | Weakly hyperbolic equations with time degeneracy in Sobolev spaces |
| title_fullStr | Weakly hyperbolic equations with time degeneracy in Sobolev spaces |
| title_full_unstemmed | Weakly hyperbolic equations with time degeneracy in Sobolev spaces |
| title_short | Weakly hyperbolic equations with time degeneracy in Sobolev spaces |
| title_sort | weakly hyperbolic equations with time degeneracy in sobolev spaces |
| topic | Quasilinear weakly hyperbolic equations time degeneracy local existence Levi conditions Sobolev spaces energy method. |
| url | http://dx.doi.org/10.1155/S1085337597000377 |
| work_keys_str_mv | AT michaelreissig weaklyhyperbolicequationswithtimedegeneracyinsobolevspaces |