Abstract
The goal of the present paper is to derive statements about energy estimates as well as L p−L q decay estimates for a Klein–Gordon model with a particular time-dependent mass. The study of this special case of a scale-invariant model is an important step within a systematic investigation of Klein–Gordon models with time-dependent mass.
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References
Abramowitz, M., Stegun, I.A. (eds.): Pocketbook of mathematical functions. Verlag Harri Deutsch, Thun (1984) [Abridged edition of Handbook of mathematical functions edited by Milton Abramowitz and Irene A. Stegun, Material selected by Michael Danos and Johann Rafelski]
Bateman H., Erdélyi A.: Higher transcendental functions, vol. 1. McGraw-Hill, New York (1953)
Böhme, C.: Decay rates and scattering states for wave models with time-dependent potential. PhD thesis, TU Bergakademie Freiberg, Germany (2011)
Böhme, C., Hirosawa, F.: Generalized energy conservation for Klein-Gordon type equations. Osaka J. Math. 49(2) (2012)
Brenner P.: On L p −L p' estimates for the wave-equation. Math. Z. 145, 251–254 (1975)
Del Santo D., Kinoshita T., Reissig M.: Klein-Gordon type equations with a singular time-dependent potential. Rendiconti dell’Istituto di Matematica dell’Universita di Trieste 39, 1–35 (2007)
Hirosawa, F., Reissig, M.: From wave to Klein-Gordon type decay rates. In: Nonlinear hyperbolic equations, spectral theory, and wavelet transformations. Operator Theory Advances and Applications, vol. 145, Birkhäuser, Basel, pp. 95–155 (2003)
Hörmander L.: Estimates for translation invariant operators in l p spaces. Acta Math. 104, 93–140 (1960)
Pecher H.: L p-Abschätzungen und klassische Lösungen für nichtlineare Wellengleichungen. I. Math. Z. 150, 159–183 (1976)
Ruzhansky M., Wirth J.: Dispersive estimates for hyperbolic systems with time-dependent coefficients. J. Differ. Equ. 251(2), 941–969 (2011)
Strichartz R.S.: Convolutions with kernels having singularities of a sphere. Trans. Am. Math. Soc. 148, 461–471 (1970)
von Wahl W.: L p-decay rates for homogeneous wave-equations. Math. Z. 120, 93–106 (1971)
Watson, G.N.: A treatise on the theory of Bessel functions, 2nd edn. Cambridge Mathematical Library. Cambridge University Press, Cambridge, vi, p. 804 (1995)
Wirth J.: Solution representations for a wave equation with weak dissipation. Math. Methods Appl. Sci. 27(1), 101–124 (2004)
Wirth J.: Wave equations with time-dependent dissipation. I: Non-effective dissipation. J. Differ. Equ. 222(2), 487–514 (2006)
Wirth J.: Wave equations with time-dependent dissipation. II: Effective dissipation. J. Differ. Equ. 232(1), 74–103 (2007)
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Böhme, C., Reissig, M. A scale-invariant Klein–Gordon model with time-dependent potential. Ann Univ Ferrara 58, 229–250 (2012). https://doi.org/10.1007/s11565-012-0153-9
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DOI: https://doi.org/10.1007/s11565-012-0153-9