Klein-Gordon type decay rates for wave equations with time-dependent coefficients

Michael Reissig; Karen Yagdjian

Banach Center Publications (2000)

  • Volume: 52, Issue: 1, page 189-212
  • ISSN: 0137-6934

Abstract

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This work is concerned with the proof of L p - L q decay estimates for solutions of the Cauchy problem for the Klein-Gordon type equation u t t - λ 2 ( t ) b 2 ( t ) ( Δ u - m 2 u ) = 0 . The coefficient consists of an increasing smooth function λ and an oscillating smooth and bounded function b which are uniformly separated from zero. Moreover, m 2 is a positive constant. We study under which assumptions for λ and b one can expect as an essential part of the decay rate the classical Klein-Gordon decay rate n/2(1/p-1/q).

How to cite

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Reissig, Michael, and Yagdjian, Karen. "Klein-Gordon type decay rates for wave equations with time-dependent coefficients." Banach Center Publications 52.1 (2000): 189-212. <http://eudml.org/doc/209057>.

@article{Reissig2000,
abstract = {This work is concerned with the proof of $L_p - L_q$ decay estimates for solutions of the Cauchy problem for the Klein-Gordon type equation $u_\{tt\} - λ^2(t)b^2(t) (Δu - m^\{2\}u) = 0$. The coefficient consists of an increasing smooth function $λ$ and an oscillating smooth and bounded function b which are uniformly separated from zero. Moreover, $m^2$ is a positive constant. We study under which assumptions for λ and b one can expect as an essential part of the decay rate the classical Klein-Gordon decay rate n/2(1/p-1/q).},
author = {Reissig, Michael, Yagdjian, Karen},
journal = {Banach Center Publications},
keywords = {- decay estimates},
language = {eng},
number = {1},
pages = {189-212},
title = {Klein-Gordon type decay rates for wave equations with time-dependent coefficients},
url = {http://eudml.org/doc/209057},
volume = {52},
year = {2000},
}

TY - JOUR
AU - Reissig, Michael
AU - Yagdjian, Karen
TI - Klein-Gordon type decay rates for wave equations with time-dependent coefficients
JO - Banach Center Publications
PY - 2000
VL - 52
IS - 1
SP - 189
EP - 212
AB - This work is concerned with the proof of $L_p - L_q$ decay estimates for solutions of the Cauchy problem for the Klein-Gordon type equation $u_{tt} - λ^2(t)b^2(t) (Δu - m^{2}u) = 0$. The coefficient consists of an increasing smooth function $λ$ and an oscillating smooth and bounded function b which are uniformly separated from zero. Moreover, $m^2$ is a positive constant. We study under which assumptions for λ and b one can expect as an essential part of the decay rate the classical Klein-Gordon decay rate n/2(1/p-1/q).
LA - eng
KW - - decay estimates
UR - http://eudml.org/doc/209057
ER -

References

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  14. [14] M. Reissig and K. Yagdjian, L p - L q estimates for the solutions of strictly hyperbolic equations of second order with time dependent coefficients, accepted for publication in Mathematische Nachrichten. 
  15. [15] M. Reissig and K. Yagdjian, L p - L q estimates for the solutions of hyperbolic equations of second order with time dependent coefficients - Oscillations via growth -, Preprint 98-5, Fakultät für Mathematik und Informatik, TU Bergakademie Freiberg. 
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  18. [18] K. Yagdjian, The Cauchy problem for hyperbolic operators. Multiple characteristics. Micro-local approach, Math. Topics, Vol. 12, Akademie Verlag, Berlin, 1997. Zbl0887.35002

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