Dispersive and Strichartz Estimates for Hyperbolic Equations with Constant Coefficients

Ruzhansky, Michael; Smith, J. G.

doi:10.1142/e022

Cited by 16 publications

(17 citation statements)

References 29 publications

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“…In this case, even in the situation of the lower regularity of coefficients (C 1 ), one can analyse the global behaviour of solutions with respect to time (see [14]). The cases of constant coefficients and systems with controlled oscillations have been treated in [17,18], respectively.…”

Section: Theorem 1 ([13]) Assume That N = 1 and That The Differentialmentioning

confidence: 99%

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

Garetto

Ruzhansky

2013

Math. Ann.

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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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Section: Theorem 1 ([13]) Assume That N = 1 and That The Differentialmentioning

confidence: 99%

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

Garetto

Ruzhansky

2013

Math. Ann.

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“…Large hyperbolic systems appear in many applications, for example the Grad systems of gas dynamics, hyperbolic systems in the Hermite‐Grad decomposition of the Fokker‐Planck equation, etc. Thus, for general hyperbolic equations with constant coefficients a comprehensive analysis of dispersive and Strichartz estimates was carried out in 14. The dispersion for scalar equations based on the asymptotic integration method was analysed by the authors in 8, motivated by the higher order Kirchhoff equations.…”

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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“…For instance, it can provide a basis to derive Strichartz decay estimates [15]. We refer the interested reader to [14] for dispersive and Strichartz estimates for solutions of higher order equations with constant coefficients.…”

Section: Introductionmentioning

confidence: 99%

Hyperbolic-like estimates for higher order equations

D’Abbicco

Ebert

2012

Journal of Mathematical Analysis and Applications

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a b s t r a c tThe main goal of this paper is to derive long time estimates of the energy for the higher order hyperbolic equations with time-dependent coefficients. In particular, we estimate the energy in the hyperbolic zone of the extended phase space by means of a function f (t) which depends on the principal part and on the coefficients of the terms of order m−1. Then we look for sufficient conditions that guarantee the same energy estimate from above in all the extended phase space. We call this class of estimates hyperbolic-like since the energy behavior is deeply depending on the hyperbolic structure of the equation. In some cases, these estimates produce a dissipative effect on the energy.

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Dispersive and Strichartz Estimates for Hyperbolic Equations with Constant Coefficients

Cited by 16 publications

References 29 publications

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

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

Scattering for strictly hyperbolic systems with time‐dependent coefficients

Hyperbolic-like estimates for higher order equations

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