Abstract
The goal of the paper is to derive L p —L q decay estimates for Klein—Gordon equations with time-dependent coefficients. We explain the influence of the relation between the mass term and the wave propagation speed on L p —L q decay estimates. Contrary to the classical Klein—Gordon case we cannot expect in each case a Klein—Gordon type decay rate. One has wave type decay rates, too. Moreover, under certain assumptions no L p —L q decay estimates can be proved. In these cases the solution has a Floquet behavior. More precisely, one can show that the energy cannot be estimated from above by time-dependent functions with a suitable growth order if t tends to infinity.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
P. Brenner, On L p —L q estimates for the wave-equation, Math. Zeit. 145 (1975), no. 3, 251–254.
F. Colombini, D. DelSanto and M. Reissig, On the optimal regularity of coefficients in hyperbolic Cauchy problems,Preprint del Dipartimento di Scienze matematiche dell’Universita di Trieste, n. 517 (2002).
M.S.P. Eastham, The Spectral Theory of Periodic Differential Equations,Scottish Academic Press, Edinburgh and London, 1973.
A. Galstian and M. Reissig, L p — L q decay estimates for a Klein-Gordon type model equation, Eds. H. Begehr et al.,Proceedings of the second ISAAC congress, vol. 2, 1355–1369, Kluwer (2000).
F. Hirosawa, Energy decay for degenerate hyperbolic equations of Klein-Gordon type with dissipative term, Funkcial. Ekvac. 43 (2000), no. 1, 163–191.
F. Hirosawa, On the Cauchy problem for second order strictly hyperbolic equations with non-regular coefficients, to appear.
F. Hirosawa, Precise energy decay estimates for the dissipative hyperbolic equations in an exterior domain,Mathematical Research Note 2000–008, Institute of Mathematics, University of Tsukuba.
L. Hörmander, Translation invariant operators in L P space,Acta Math. 104 (1960), 93–140.
L. Hörmander, Remarks on the Klein-Gordon equation, Journées Equations aux Dérivées Partielles, Saint Jean de Monts, 1987.
S. Kawashima, M. Nakao and K. Ono, On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term,J. Math. Soc. Japan 47 (1995), no. 4, 617–653.
S. Klainerman, Global existence for nonlinear wave equations,Comm. Pure Appl. Math. 33 (1980), no. 1, 43–101.
A. Kubo and M. Reissig, Construction of parametrix for hyperbolic equations with fast oscillations in non-Lipschitz coefficients, Mathematical Research Note 2002–003, Institute of Mathematics, University of Tsukuba.
O. Liess, Decay estimates for the solutions of the system of crystal optics, Asymptotic Analysis 4 (1991), no. 1, 61–95.
W. Littman, Fourier transformations of surface carried measures and differentiability of surface averages, Bull. Amer. Math. Soc. 69 (1963), 766–770.
W. Magnus and St. Winkler, Hill’s Equation,Interscience Publishers, New YorkLondon-Sydney, 1966.
A. Matsumura, Energy decay of solutions of dissipative wave equations, Proc. Japan Acad. Ser. A Math. Sci. 53 (1977), no. 7, 232–236.
K. Mochizuki and T. Motai, On energy decay-nondecay problems for wave equations with nonlinear dissipative term in \( {\mathbb{R}^N} \) J. Math. Soc. Japan 47 (1995), no. 3, 405–421.
H. Pecher, L p -Abschätzungen and klassische Lösungen für nichtlineare Wellenglei chungen. I,Math. Zeitschrift 150 (1976), no. 2, 159–183.
R. Racke, Lectures on nonlinear evolution equations,Aspects of Mathematics, Vieweg, Braunschweig/Wiesbaden, 1992.
M. Reissig, Klein-Gordon type decay rates for wave equations with a time-dependent dissipation,Adv. Math. Sci. Appl. 11 (2001), no. 2, 859–891.
M. Reissig and K. Yagdjian, About the influence of oscillations on Strichartz-type decay estimates, Rend. Sem. Mat. Univ. Polit. Torino 58 (2000), no. 3, 117–130.
M. Reissig and K. Yagdjian, One application of Floquet’s theory to L p — L q estimates for hyperbolic equations with very fast oscillations,Math. Meth. Appl. Sci. 22 (1999), no. 11, 937–951.
M. Reissig and K. Yagdjian, Klein-Gordon type decay rates for wave equations with time-dependent coefficients,Banach Center Publications, 52 (2000), 189–212.
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 1998, ISSN 1433–9307.
M. Reissig and K. Yagdjian, Weakly hyperbolic equations with fast oscillating coefficients,Osaka J. Math. 36 (1999), no. 2, 437–464.
M. Stoth, Globale klassische Lösungen der quasilinearen Elastizitätsgleichungen für kubisch elastische Medien im \(\mathbb{R}^2 \),SFB 256 Preprint 157, Universität Bonn, 1991.
R. Strichartz, A priori estimates for the wave-equation and some applications,J. Funct. Anal. 5 (1970), 218–235.
H. Uesaka, The total energy decay of solutions for the wave equation with a dissipative term,J. Math. Kyoto Univ. 20 (1980), no. 1, 57–65.
W. v. Wahl, L P -decay rates for homogeneous wave-equations,Math. Zeit. 120 (1971), 93–106.
K. Yagdjian, The Cauchy problem for hyperbolic operators. Multiple characteristics,micro-local approach, Math. Topics, Akademie-Verlag, Berlin, 1997.
K. Yagdjian, Parametric resonance and nonexistence of global solution to nonlinear wave equations, J. Math. Anal. Appl. 260 (2001), no. 1, 251–268.
S. Zheng and W. Shen, Global solutions to the Cauchy problem of quasilinear hyperbolic parabolic coupled systems, Sci. Sinica, Ser. A 30 (1987), no. 11, 1133–1149.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Basel AG
About this chapter
Cite this chapter
Hirosawa, F., Reissig, M. (2003). From Wave to Klein—Gordon Type Decay Rates. In: Albeverio, S., Demuth, M., Schrohe, E., Schulze, BW. (eds) Nonlinear Hyperbolic Equations, Spectral Theory, and Wavelet Transformations. Operator Theory: Advances and Applications, vol 145. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8073-2_2
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8073-2_2
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9429-6
Online ISBN: 978-3-0348-8073-2
eBook Packages: Springer Book Archive