Glasnik Matematicki, Vol. 54, No. 1 (2019), 179-209.

TIME DECAY ESTIMATES FOR WAVE EQUATIONS WITH TRANSMISSION AND BOUNDARY CONDITIONS

Krešo Mihalinčić

Faculty of Tourism and Hospitality Management, University of Rijeka, Primorska 42, 51410 Opatija, Croatia
e-mail: kresom@fthm.hr

Abstract.   Time decay estimates are derived for solutions of some initial value problems of wave propagation, based on the method of stationary phase. Solutions to three dimensional wave equation in wedges and one dimensional wave equation with a constant potential are shown to decay like t-1 and t-1/2, respectively. Dependencies of the results on initial data and physical implications are discussed.

2010 Mathematics Subject Classification.   35L05, 74S25, 35G16

Key words and phrases.   Partial differential equations, initial boundary value problem, mathematical physics, wave propagation, stationary phase, time decay

Full text (PDF) (free access)

https://doi.org/10.3336/gm.54.1.08

References:

  1. A. Adams, Sobolev spaces, Academic Press, New York, London, 1975.
    MathSciNet    

  2. H.-D. Alber and R. Leis, Initial boundary value and scattering problems in mathematical physics, in: Partial differential equations and calculus of variations, Springer, Berlin, 1988, 23-60.
    MathSciNet     CrossRef

  3. F. Ali Mehmeti, Regular solutions of transmission and interaction problems for wave equations, Math. Methods Appl. Sci. 11 (1989), 665-685.
    MathSciNet     CrossRef

  4. F. Ali Mehmeti, Spectral theory and L∞-time decay estimates for Klein-Gordon equations on two half axes with transmission: the tunnel effect, Math. Methods Appl. Sci. 17 (1994), 697-752.
    MathSciNet     CrossRef

  5. F. Ali Mehmeti, R. Haller-Dintelmann and V. Regnier, The influence of the tunnel effect on L∞-time decay, in: Spectral theory, mathematical system theory, evolution equations, differential and difference equations. Birkhäuser, Basel, 2012, 11-24.
    MathSciNet     CrossRef

  6. F. Ali Mehmeti, E. Meister and K. Mihalinčić, Spectral theory for the wave equation in two adjacent wedges, Math. Methods Appl. Sci. 20 (1997), 1015-1044.
    MathSciNet     CrossRef

  7. L. Aloui, S. Ibrahim and K. Nakanishi , Exponential energy decay for damped Klein-Gordon equation with nonlinearities of arbitrary growth, Comm. Partial Differential Equations 36 (2011), 797-818.
    MathSciNet     CrossRef

  8. D. Andrade, L.H. Fatori and J.E. Muñoz Rivera, Nonlinear transmission problem with a dissipative boundary condition of memory type, Electron. J. Differential Equations 2006, No. 53, 16 pp.
    MathSciNet    

  9. M. Beals and W. Strauss, Lp-estimates for the wave equation with a potential, Comm. Partial Differential Equations 18 (1993), 1365-1397.
    MathSciNet     CrossRef

  10. M. Daoulatli, Energy decay rates for solutions of the wave equation with linear damping in exterior domain, Evol. Equ. Control Theory 5 (2016), 37-59.
    MathSciNet     CrossRef

  11. M. Daoulatli, Energy decay rates for solutions of the wave equations with nonlinear damping in exterior domain, J. Differential Equations 264 (2017), 4260-4302.
    MathSciNet     CrossRef

  12. L. Hörmander, The analysis of linear partial differential operators. I., Springer-Verlag, Berlin, 1983.
    MathSciNet     CrossRef

  13. J.B. Jeong, Nonlinear transmission problem for wave equation with boundary condition of memory type, Acta Appl. Math. 110 (2010), 907-919.
    MathSciNet     CrossRef

  14. F. John, Existence for large times of strict solutions of nonlinear wave equations in three space dimensions for small initial data, Comm. Pure Appl. Math. 40 (1987), 79-109.
    MathSciNet     CrossRef

  15. F. John and S. Klainerman, Almost global existence to nonlinear wave equations in three space dimensions, Comm. Pure Appl. Math. 37 (1984), 443-455.
    MathSciNet     CrossRef

  16. S. Klainerman and G. Ponce, Global, small amplitude solutions to nonlinear evolution equations, Comm. Pure Appl. Math. 36 (1983), 133-141.
    MathSciNet     CrossRef

  17. R. Leis, Initial boundary value problems in mathematical physics, B.G. Teubner, Stuttgart, J. Wiley, Chichester, 1986.
    MathSciNet     CrossRef

  18. O. Liess, Decay estimates for the solutions of the system of crystal optics, Asymptotic Anal. 4 (1991), 61-95.
    MathSciNet    

  19. B. Marshall, W. Strauss and S. Wainger, Lp-Lq estimates for the Klein-Gordon equation, J. Math. Pures Appl. 59 (1980), 417-440.
    MathSciNet    

  20. E. Meister, Some solved and unsolved canonical transmission problems of diffraction theory, in: Problems and methods in mathematical physics, Teubner, Stuttgart, 1994., 89-99.
    MathSciNet    

  21. K. Mihalinčić, Time decay estimates for the wave equation with transmission and boundary conditions, Dissertation, Technische Universität Darmstadt, Germany, 1998.

  22. K. Mihalinčić, Wave equation in halfspace and in two adjacent wedges, ZAMM Z. Angew. Math. Mech. 78 (1998) S3, 1017-1018.
    CrossRef

  23. R. Racke, Lectures on nonlinear evolution equations, Friedr. Vieweg & Sohn, Braunschweig, 1992.
    MathSciNet     CrossRef

  24. M. Reed and B. Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York and London, 1972.
    MathSciNet    

  25. W. von Wahl, Lp- decay rates for homogeneous wave-equations, Math. Z. 120 (1971), 93-106.
    MathSciNet     CrossRef

  26. P. Werner, Resonance phenomena in infinite strings with periodic ends, Math. Methods Appl. Sci. 17 (1994), 115-154.
    MathSciNet     CrossRef

  27. C.H. Wilcox, Sound propagation in stratified fluids, Springer-Verlag, New York, 1984.
    MathSciNet     CrossRef

  28. E.C. Zachmanoglou, The decay of solutions of the initial-boundary value problem for the wave equation in unbounded regions, Arch. Rational Mech. Anal. 14 (1963), 312-325.
    MathSciNet     CrossRef

  29. A.H. Zemanian, Distribution theory and transform analysis. An introduction to generalized functions, with applications, McGraw-Hill, New York-Toronto-London-Sydney, 1965.
    MathSciNet    

  30. W.R. Green, Time decay estimates for the wave equation with potential in dimension two, J. Differential Equations 257 (2014), 868-919.
    MathSciNet     CrossRef
Glasnik Matematicki Home Page