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

Michael Reissig; Karen Yagdjian

ArticleOriginal scientific text

Title

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

Authors ¹, ²

Affiliations

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

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) (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).

P. Brenner, On $L_{p} - L_{q}$ estimates for the wave-equation, Math. Zeitschrift 145 (1975), 251-254.
P. Brenner, On the existence of global smooth solutions of certain semi-linear hyperbolic equations, Math. Zeitschrift 167 (1979), 99-135.
A. Grigis and J. Sjöstrand, Microlocal Analysis for Differential Operators. An Introduction, London Mathematical Society Lecture Note Series, Vol.196, University Press, Cambridge, 1994.
F. Hirosawa, Energy decay for degenerate hyperbolic equations of Klein-Gorden type with dissipative term, manuscript.
L. Hörmander, Remarks on the Klein-Gorden equation, Journées Equations aux Dérivées Partielles, Saint Jean de Monts, 1987.
L. Hörmander, Translation invariant operators in $L^{p}$ spaces, Acta Math. 104 (1960), 93-140.
S. Klainerman, Global existence for nonlinear wave equations, Comm. on Pure and Appl. Math. 33 (1980), 43-101.
Li Ta-tsien, Global classical solutions for quasilinear hyperbolic systems, John Wiley & Sons, 1994.
W. Littman, Fourier transformations of surface carried measures and differentiability of surface averages, Bull. Amer. Math. Soc. 69 (1963), 766-770.
H. Pecher, $L_{p}$ -Abschätzungen und klassische Lösungen für nichtlineare Wellengleichungen. I, Math. Zeitschrift 150 (1976), 159-183.
R. Racke, Lectures on Nonlinear Evolution Equations, Aspects of Mathematics, Vieweg, Braunschweig/Wiesbaden, 1992.
M. Reissig and K. Yagdjian, An example for the influence of oscillations on $L_{p} - L_{q}$ decay estimates, to appear.
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.
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.
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.
R. Strichartz, A priori estimates for the wave-equation and some applications, J. Funct. Anal. 5 (1970), 218-235.
W. v. Wahl, $L^{p}$ -decay rates for homogeneous wave-equations, Math. Zeitschrift 120 (1971), 93-106.
K. Yagdjian, The Cauchy problem for hyperbolic operators. Multiple characteristics. Micro-local approach, Math. Topics, Vol. 12, Akademie Verlag, Berlin, 1997.

Title

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

Affiliations

Abstract

Bibliography