The Cauchy Problem for Strictly Hyperbolic Operators with Non-Absolutely Continuous Coefficients

Cicognani, Massimo

doi:10.21099/tkbjm/1496164556

Cited by 27 publications

(37 citation statements)

References 2 publications

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“…Following [4,5], we show that Theorem 2.1 implies the H ±∞ well-posedness of problem (4.1) also in this case. Then the Cauchy problem (4.1) is well-posed in H ±∞ .…”

Section: Strictly Hyperbolic Equationsmentioning

confidence: 65%

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Energy estimate and fundamental solution for degenerate hyperbolic Cauchy problems

Ascanelli¹,

Cicognani²

2005

Journal of Differential Equations

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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 ∞ Levi condition, and a strictly hyperbolic equation with non-regular coefficients with respect to the time variable can be reduced to firstorder 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 ∞ . 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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“…Following [4,5], we show that Theorem 2.1 implies the H ±∞ well-posedness of problem (4.1) also in this case. Then the Cauchy problem (4.1) is well-posed in H ±∞ .…”

Section: Strictly Hyperbolic Equationsmentioning

confidence: 65%

“…The optimal exponent for the C ∞ well-posedness is q = 1. The dependence on space variables was allowed in [4], the sharp bound…”

Section: Introductionmentioning

confidence: 99%

Energy estimate and fundamental solution for degenerate hyperbolic Cauchy problems

Ascanelli¹,

Cicognani²

2005

Journal of Differential Equations

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“…These results have been improved by Cicognani [3], [4] allowing dependence of the coefficients on x and equations of higher order .…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

A degenerate hyperbolic equation under Levi conditions

Ascanelli

2006

Ann. Univ. Ferrara

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We consider the Cauchy problem for a second order equation of hyperbolic type. This equation degenerates in two different ways. On one hand, the coefficients have a bad behavior with respect to time: there is a blow-up phenomenon in the first time derivative of the principal part's coefficients, that is the derivative vanishes at the time t = 0. On the other hand, the equation is weakly hyperbolic and the multiplicity of the roots is not constant, but zeroes are of finite order.Here we overcome the blow-up problem and, moreover, the finitely degeneration of the Cauchy problem allows us to give an appropriate Levi condition on the lower order terms in order to get C ∞ well posedness of the Cauchy problem.

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“…Thus if we are interested in H ∞ well-posedness results for higher order equations, then we are not able to formulate assumptions as (2.4) with a 1, but we are forced to choose a = 0. This is done in [1,2].…”

Section: Relations To Previous Resultsmentioning

confidence: 99%