On the Cauchy Problem for Hyperbolic Operators with Non-Regular Coefficients

Colombini, Ferruccio; Santo, Daniele Del; Kinoshita, Tamotu

doi:10.1007/978-94-017-2008-3_3

Cited by 34 publications

(66 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For example, after regularising (for example non-Lipschitz, Hölder, etc.) coefficients with a parameter ε, relating ε to some frequency zones in the energy estimate often yields the Gevrey or even C ∞ well-posedness (see for example [6,7], and other papers). It is not always possible to relate ε to frequency zones in which case families of solutions can be considered as a whole: for example, for hyperbolic equations with discontinuous coefficients, regularised families have been already considered by Hurd and Sattinger [21], with a subsequent analysis of limits of these regularisations in L 2 as ε → 0.…”

Section: Introductionmentioning

confidence: 99%

Hyperbolic Second Order Equations with Non-Regular Time Dependent Coefficients

Garetto

Ruzhansky

2014

Arch Rational Mech Anal

113

View full text Add to dashboard Cite

In this paper we study weakly hyperbolic second order equations with time dependent irregular coefficients. This means assuming that the coefficients are less regular than Hölder. The characteristic roots are also allowed to have multiplicities. For such equations, we describe the notion of a 'very weak solution' adapted to the type of solutions that exist for regular coefficients. The construction is based on considering Friedrichs-type mollifiers of coefficients and corresponding classical solutions, and their quantitative behaviour in the regularising parameter. We show that even for distributional coefficients, the Cauchy problem does have a very weak solution, and that this notion leads to classical or to ultradistributional solutions under conditions when such solutions also exist. In concrete applications, the dependence on the regularising parameter can be traced explicitly.

show abstract

Section: Introductionmentioning

confidence: 99%

Hyperbolic Second Order Equations with Non-Regular Time Dependent Coefficients

Garetto

Ruzhansky

2014

Arch Rational Mech Anal

113

View full text Add to dashboard Cite

show abstract

“…This means that some singular behavior of the coefficient with respect to the Lipschitz continuity brings loss of regularity of the solution. We are referring to the following well known result: It is also considered in [3] the mixed case of the conditions (1.2) and (1.4) as follows:…”

Section: Theorem 12 ([5] (Cf [7])) Assume That A(t) Satisfiesmentioning

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.

View full text Add to dashboard Cite

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.

show abstract

“…From [3] (see also [2]) we obtain the well-posedness in C ∞ . This concludes the proof of Theorem 1.4.…”

Section: Lemma 31 Let 0 < ε ≤ 1 and ρ(T) Be A Non-negative Functionmentioning

confidence: 99%

“…It is then natural to introduce, apart from the regularity, a new condition on the coefficients in order to control their oscillations. In our previous papers [2], [3] and [4] we assumed the oscillation condition as follows: if q ≥ 3 Remark 1.1. As the parameter q tends to infinity, the oscillation condition (1.6) loses sense and (1.7) and (1.8) approach to the results of [1] and [6].…”

Section: Introductionmentioning

confidence: 99%

“…Paragraph 3 is devoted to the proof of the Theorem 1.4 and Paragraph 4 to that one of the Theorem 1.7. In the last paragraph we give a sketch of the proof of the Theorem 1.11 and only some ideas about that one of the Theorem 1.12, pointing out to the interested reader the similar constructions of counter examples in [1], [6], [7], [2], [3], [4] and [5].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation