An example of a weakly hyperbolic Cauchy problem not well posed in C∞

In this paper we consider weakly hyperbolic equations of higher orders in arbitrary dimensions with time-dependent coefficients and lower order terms. We prove the Gevrey well-posedness of the Cauchy problem under C k -regularity of coefficients of the principal part and natural Levi conditions on lower order terms which may be only continuous. In the case of analytic coefficients in the principal part we establish the C ∞ well-posedness. The proofs are based on using the quasi-symmetriser for the corresponding companion system and inductions on the order of equation and on the frequency regions. The main novelty compared to the existing literature is the possibility to include lower order terms to the equation (which have been untreatable until now in these problems) as well as considering any space dimensions. We also give results on the ultradistributional and distributional well-posedness of the problem, and we look at new effects for equations with discontinuous lower order terms.

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“…By comparison with the Levi conditions (7) and the identity (8), it is clear that we impose conditions on less terms of the equation.…”

Section: To Be the Zeros Of F (· ξ)mentioning

confidence: 99%

“…On the other hand, in the celebrated paper [8], Colombini and Spagnolo constructed a C ∞ function a(t) ≥ 0 such that (5) is not C ∞ well-posed. The situation becomes even more complicated if one adds mixed terms to (5), even depending only on t and analytic.…”

Section: Introductionmentioning

confidence: 99%

Weakly hyperbolic equations with non-analytic coefficients and lower order terms

Garetto

Ruzhansky

2013

Math. Ann.

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“…, [4], [5]). Furthermore, singularities of the initial data may propagate in an astonishing way: in [14], it has been shown that the solution v = v(t, x) of This representation shows that singularities of the initial datum u 0 propagate only to the left.…”

Section: Introductionmentioning

confidence: 99%

Edge Sobolev spaces and weakly hyperbolic equations

Dreher¹,

Witt²

2002

Ann. Mat. Pura Appl. IV. Ser.

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“…It is our aim in this paper to show that the interval of Gevrey well-posedness in (5) (4). In this sense, the well-posedness of (1) can be determined by studying the well-posedness of the reduced Cauchy problem (4).…”

Section: Introductionmentioning

confidence: 99%

On hyperbolic systems with time-dependent Hölder characteristics

Garetto

Ruzhansky

2016

Annali di Matematica

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In this paper, we study the well-posedness of weakly hyperbolic systems with time-dependent coefficients. We assume that the eigenvalues are low regular, in the sense that they are Hölder with respect to t. In the past, these kinds of systems have been investigated by Yuzawa (J Differ Equ 219 (2) (2005) and we obtain well-posedness in spaces of ultradistributions as well. Our main idea is a reduction of the system to block Sylvester form and then the formulation of suitable energy estimates inspired by the treatment of scalar equations in Garetto and Ruzhansky (J Differ Equ 253(5):1317-1340, 2012).

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An example of a weakly hyperbolic Cauchy problem not well posed in C∞

Cited by 113 publications

References 1 publication

Weakly hyperbolic equations with non-analytic coefficients and lower order terms

Weakly hyperbolic equations with non-analytic coefficients and lower order terms

Edge Sobolev spaces and weakly hyperbolic equations

On hyperbolic systems with time-dependent Hölder characteristics

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