Asymptotic behaviour for wave equation with time-dependent coefficients

Matsuyama, Tokio

doi:10.1007/s11565-006-0028-z

Cited by 8 publications

(19 citation statements)

References 4 publications

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“…We note that for the second order equations we have m = 2 and the next theorem covers the case of the wave equation as a special case, also improving the corresponding result in [13,14,15]. The result is as follows:…”

Section: Representation Of Solutions To Linear Cauchy Problemssupporting

confidence: 73%

Dispersion and Asymptotic Profiles for Kirchhoff Equations

Matsuyama¹,

Ruzhansky²

2007

Topics in Contemporary Differential Geometry, Complex Analysis and Mathematical Physics

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Abstract. The aim of this article is to describe asymptotic profiles for the Kirchhoff equation, and to establish time decay properties and dispersive estimates for Kirchhoff equations. For this purpose, the method of asymptotic integration is developed for the corresponding linear equations and representation formulae for their solutions are obtained. These formulae are analysed further to obtain the time decay rate of L p -L q norms of propagators for the corresponding Cauchy problems.

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Section: Representation Of Solutions To Linear Cauchy Problemssupporting

confidence: 73%

Dispersion and Asymptotic Profiles for Kirchhoff Equations

Matsuyama¹,

Ruzhansky²

2007

Topics in Contemporary Differential Geometry, Complex Analysis and Mathematical Physics

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“…The following results extend some of those in [2][3][4][5][6] and [15] to the setting of higher order equations, at the same time improving the indices in regularity assumptions. Full analysis will appear in [7], where we also develop the asymptotic integration methods in the PDE context (for the ODE setting see e.g.…”

Section: Nonlinear Equations Of Kirchhoff Typesupporting

confidence: 62%

Time decay for hyperbolic equations with homogeneous symbols

Matsuyama

Ruzhansky

2009

Comptes Rendus. Mathématique

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The aim of this Note is to present dispersive estimates for strictly hyperbolic equations with time dependent coefficients that have integrable derivatives. We will relate the time decay rate of L p -L q norms of solutions to certain geometric indices associated to the characteristics of the limiting equation. Results will be applied to the global solvability of Kirchhoff type equations with small data, and to the dispersive estimates for their solutions. To cite this article: T. Matsuyama, M. Ruzhansky, C. R. Acad. Sci. Paris, Ser. I 347 (2009). Résumé Estimations en temps pour des équations hyperboliques à symboles homogènes. Dans cette Note, nous établissons des estimations dispersives des solutions d'équations strictement hyperboliques à coefficients dépendant du temps et à dérivées intégrables. Nous estimons le taux de décroissance en temps des normes des solutions en fonction d'indices géométriques associés aux caractéristiques de l'équation limite. Les résultats sont appliqués à la résolvabilité globale des équations de type Kirchhoff à données petites et à des estimations dispersives des solutions. Pour citer cet article : T. Matsuyama, M. Ruzhansky, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

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“…As to the strictly hyperbolic equations of second order for “bounded domains,” a similar result was obtained in 1. It should be noted that the results of 5 are applied to deduce non‐existence of scattering states for the Kirchhoff equation (see 6). In this sense the behaviour of c ( t ) − c ±∞ affects the development of scattering theory for wave equations with time‐dependent coefficients as well as for the Kirchhoff equation.…”

Section: Introductionmentioning

confidence: 56%

“…The important feature for these results is that we limit the conditions on taking only one derivative of the coefficients keeping it in the framework of the study of asymptotic behaviours for Kirchhoff systems. In 5 the first author gave a sufficient condition on the existence of scattering states for wave equations, and found a special data for non‐existence of scattering states. More precisely, there exists a solution u = u ( t , x ) of the Cauchy problem to strictly hyperbolic equation of second order of the form such that u is not asymptotically free, where we assume that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$c(t)\in \mathrm{Lip}_{\mathrm{loc}}({\mathbb {R}})$\end{document} satisfies and the improper Riemann integrals of c ( t ) − c ±∞ do not exist.…”

Section: Introductionmentioning

confidence: 99%

Scattering for strictly hyperbolic systems with time‐dependent coefficients

Matsuyama

Ruzhansky

2013

Mathematische Nachrichten

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The present paper is devoted to finding conditions on the occurrence of scattering for strictly hyperbolic systems with time-dependent coefficients whose time-derivatives are in L 1 in time. More precisely, it will be shown that the solutions are asymptotically free if the coefficients are stable in the sense that their improper Riemann integrals converge as t → ±∞, while each nontrivial solution with radially symmetric data is never asymptotically free provided that the coefficients are not stable as t → ±∞. As a by-product, wave and scattering operators can be constructed. An important feature is that assumptions on only one derivative of the coefficients are made so that the results would be applicable to the asymptotic behaviour of Kirchhoff systems.

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Asymptotic behaviour for wave equation with time-dependent coefficients

Cited by 8 publications

References 4 publications

Dispersion and Asymptotic Profiles for Kirchhoff Equations

Dispersion and Asymptotic Profiles for Kirchhoff Equations

Time decay for hyperbolic equations with homogeneous symbols

Scattering for strictly hyperbolic systems with time‐dependent coefficients

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