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2000 | 52 | 1 | 189-212
Tytuł artykułu

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

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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).
Słowa kluczowe
Rocznik
Tom
52
Numer
1
Strony
189-212
Opis fizyczny
Daty
wydano
2000
Twórcy
  • Faculty of Mathematics and Computer Science, Technical University Bergakademie Freiberg, Bernhard von Cotta Str. 2, 09596 Freiberg, Germany
  • Institute of Mathematics, University of Tsukuba , Tsukuba, Ibaraki 305, Japan
Bibliografia
  • [1] P. Brenner, On $L_p-L_q$ estimates for the wave-equation, Math. Zeitschrift 145 (1975), 251-254.
  • [2] P. Brenner, On the existence of global smooth solutions of certain semi-linear hyperbolic equations, Math. Zeitschrift 167 (1979), 99-135.
  • [3] A. Grigis and J. Sjöstrand, Microlocal Analysis for Differential Operators. An Introduction, London Mathematical Society Lecture Note Series, Vol.196, University Press, Cambridge, 1994.
  • [4] F. Hirosawa, Energy decay for degenerate hyperbolic equations of Klein-Gorden type with dissipative term, manuscript.
  • [5] L. Hörmander, Remarks on the Klein-Gorden equation, Journées Equations aux Dérivées Partielles, Saint Jean de Monts, 1987.
  • [6] L. Hörmander, Translation invariant operators in $L^p$ spaces, Acta Math. 104 (1960), 93-140.
  • [7] S. Klainerman, Global existence for nonlinear wave equations, Comm. on Pure and Appl. Math. 33 (1980), 43-101.
  • [8] Li Ta-tsien, Global classical solutions for quasilinear hyperbolic systems, John Wiley & Sons, 1994.
  • [9] W. Littman, Fourier transformations of surface carried measures and differentiability of surface averages, Bull. Amer. Math. Soc. 69 (1963), 766-770.
  • [10] H. Pecher, $L_p$-Abschätzungen und klassische Lösungen für nichtlineare Wellengleichungen. I, Math. Zeitschrift 150 (1976), 159-183.
  • [11] R. Racke, Lectures on Nonlinear Evolution Equations, Aspects of Mathematics, Vieweg, Braunschweig/Wiesbaden, 1992.
  • [12] M. Reissig and K. Yagdjian, An example for the influence of oscillations on $L_p-L_q$ decay estimates, to appear.
  • [13] M. Reissig and K. Yagdjian, One application of Floquet's theory to $L_p-L_q$ estimates, Math. Meth. Appl. Sci. 22 (1999), 937-951.
  • [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] 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.
  • [16] R. Strichartz, A priori estimates for the wave-equation and some applications, J. Funct. Anal. 5 (1970), 218-235.
  • [17] W. v. Wahl, $L^p$-decay rates for homogeneous wave-equations, Math. Zeitschrift 120 (1971), 93-106.
  • [18] K. Yagdjian, The Cauchy problem for hyperbolic operators. Multiple characteristics. Micro-local approach, Math. Topics, Vol. 12, Akademie Verlag, Berlin, 1997.
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Bibliografia
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