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.

Energy estimate and fundamental solution for degnerate hyperbolic Cauchy problem

CICOGNANI, MASSIMO;
2005

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/4612
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