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EnglishThe aim of this paper is to give an uniform approach to different kinds of degenerate hyperbolic Cauchy problems. We prove that a weakly hyperbolic equation, satisfying an intermediate condition between effective hyperbolicity and the $C^{infty}$ Levi condition, and a strictly hyperbolic equation with non-regular coefficients with respect to the time variable can be reduced to first order systems of the same type. For such a kind of systems, we prove an energy estimate in Sobolev spaces (with a loss of derivatives) which gives the well-posedness of the Cauchy problem in $C^{infty}$. In the strictly hyperbolic case, we also construct the fundamental solution and we describe the propagation of the space singularities of the solution which is influenced by the non-regularity of the coefficients with respect to the time variable.
| Titolo: | Energy estimate and fundamental solution for degnerate hyperbolic Cauchy problem | |
| Autore/i: | CICOGNANI, MASSIMO; Ascanelli A. | |
| Autore/i Unibo: | ||
| Anno: | 2005 | |
| Rivista: | ||
| Digital Object Identifier (DOI): | http://dx.doi.org/10.1016/j.jde.2004.10.010 | |
| Abstract: | The aim of this paper is to give an uniform approach to different kinds of degenerate hyperbolic Cauchy problems. We prove that a weakly hyperbolic equation, satisfying an intermediate condition between effective hyperbolicity and the $C^{infty}$ Levi condition, and a strictly hyperbolic equation with non-regular coefficients with respect to the time variable can be reduced to first order systems of the same type. For such a kind of systems, we prove an energy estimate in Sobolev spaces (with a loss of derivatives) which gives the well-posedness of the Cauchy problem in $C^{infty}$. In the strictly hyperbolic case, we also construct the fundamental solution and we describe the propagation of the space singularities of the solution which is influenced by the non-regularity of the coefficients with respect to the time variable. | |
| Data prodotto definitivo in UGOV: | 2006-01-29 12:06:12 | |
| Appare nelle tipologie: | 1.01 Articolo in rivista |
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http://hdl.handle.net/11585/4612
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