Some examples of hyperbolic equations without local solvability

Colombini, Ferruccio; Spagnolo, Sergio

doi:10.24033/asens.1578

Cited by 28 publications

(22 citation statements)

References 4 publications

(5 reference statements)

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“…Our example is a radial version of the 1-dimensional example of Castro-Zuazua [3], which in turn is based on a calculation of Colombini-Spagnolo [4]. In our example the metric depends on the frequency λ (with uniform bounds on its Hölder norm), but because of the exponential localization of the eigenfunctions one may easily cut and paste together a sequence of examples to produce a metric for which these estimates fail for a sequence of λ tending to ∞, as in [3].…”

Section: Proof Of Theorems 11 and 12mentioning

confidence: 99%

Sharp $L^q$ bounds on spectral clusters for Holder metrics

Koch

Smith

Tataru

2007

Mathematical Research Letters

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Abstract. We establish L q bounds on eigenfunctions, and more generally on spectrally localized functions (spectral clusters), associated to a self-adjoint elliptic operator on a compact manifold, under the assumption that the coefficients of the operator are of regularity C s , where 0 ≤ s ≤ 1. We also produce examples which show that these bounds are best possible for the case q = ∞, and for 2 ≤ q ≤ qn.

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Section: Proof Of Theorems 11 and 12mentioning

confidence: 99%

Sharp $L^q$ bounds on spectral clusters for Holder metrics

Koch

Smith

Tataru

2007

Mathematical Research Letters

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“…Consequently, it seems to be impossible to derive any benefit from C m property with m ≥ 3 for (3) under the properties (5). However, we propose that our first expectation can be realized under an additional assumption to a(t); this is the main purpose of the present paper.…”

Section: Introductionmentioning

confidence: 76%

“…If we introduce the C ∞ property of a(t) instead of the C m property in Theorem 3, we can immediately conclude the following corollary: (5) for any k ∈ N with q ∈ [0, 1) and (6) with p ∈ [0, 1). If q > p, then the estimate (3) holds, and if q < p, then the following estimate holds:…”

Section: Introductionmentioning

confidence: 96%

On the asymptotic behavior of the energy for the wave equations with time depending coefficients

Hirosawa

2007

Math. Ann.

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We consider the asymptotic behavior of the total energy of solutions to the Cauchy problem for wave equations with time dependent propagation speed. The main purpose of this paper is that the asymptotic behavior of the total energy is dominated by the following properties of the coefficient: order of the differentiability, behavior of the derivatives as t → ∞ and stabilization of the amplitude described by an integral. Moreover, the optimality of these properties are ensured by actual examples. Mathematics Subject Classification (2000)

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“…The idea of the proof is based on the method which was developed in the series of papers [1], [2] and [4]. For v(t, ξ ) the solution of (3.1), let us define E (t, ξ ) by…”

Section: Proof Of Theorem 23mentioning

confidence: 99%

On the Gevrey well-posedness for second order strictly hyperbolic Cauchy problems under the influence of the regularity of the coefficients

Cicognani¹,

Hirosawa²

2008

MATH. SCAND.

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We consider the loss of regularity of the solution to the backward Cauchy problem for a second order strictly hyperbolic equation on the time interval [0, T ] with time depending coefficients which have a singularity only at the end point t = 0. The main purpose of this paper is to show that the loss of regularity of the solution on the Gevrey scale can be described by the order of differentiability of the coefficients on (0, T ], the order of singularities of each derivatives as t → 0 and a stabilization condition of the amplitude of oscillations described by an integral on (0, T ). Moreover, we prove the optimality of the conditions for C ∞ coefficients on (0, T ] by constructing a counterexample.

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Some examples of hyperbolic equations without local solvability

Cited by 28 publications

References 4 publications

Sharp $L^q$ bounds on spectral clusters for Holder metrics

Sharp $L^q$ bounds on spectral clusters for Holder metrics

On the asymptotic behavior of the energy for the wave equations with time depending coefficients

On the Gevrey well-posedness for second order strictly hyperbolic Cauchy problems under the influence of the regularity of the coefficients

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