Journal of the Mathematical Society of Japan

Optimal decay rate of the energy for wave equations with critical potential

Ryo IKEHATA, Grozdena TODOROVA, and Borislav YORDANOV

Full-text: Open access


We study the long time behavior of solutions of the wave equation with a variable damping term $V(x)u_t$ in the case of critical decay $V(x)\geq V_0(1+|x|^2)^{-1/2}$ (see condition (A) below). The solutions manifest a new threshold effect with respect to the size of the coefficient $V_0$: for $1 < V_0 < N$ the energy decay rate is exactly $t^{-V_0}$, while for $V_0\geq N$ the energy decay rate coincides with the decay rate of the corresponding parabolic problem.

Article information

J. Math. Soc. Japan, Volume 65, Number 1 (2013), 183-236.

First available in Project Euclid: 24 January 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35L70: Nonlinear second-order hyperbolic equations
Secondary: 35L05: Wave equation 35B33: Critical exponents 35B40: Asymptotic behavior of solutions

damped wave equation critical potential energy decay finite speed of propagation diffusive structure


IKEHATA, Ryo; TODOROVA, Grozdena; YORDANOV, Borislav. Optimal decay rate of the energy for wave equations with critical potential. J. Math. Soc. Japan 65 (2013), no. 1, 183--236. doi:10.2969/jmsj/06510183.

Export citation


  • E. B. Davies and A. M. Hinz, Explicit constants for Rellich inequalities in $L_{p}(\Omega)$, Math. Z., 227 (1998), 522–523.
  • M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation, Nonlinear Anal., 11 (1987), 1103–1133.
  • R. Ikehata, Improved decay rates for solutions to one-dimensional linear and semilinear dissipative wave equations in all space, J. Math. Anal. Appl., 277 (2003), 555–570.
  • R. Ikehata, Fast decay of solutions for linear wave equations with dissipation localized near infinity in an exterior domain, J. Differential Equations, 188 (2003), 390–405.
  • R. Ikehata, Some remarks on the wave equation with potential type damping coefficients, Int. J. Pure Appl. Math., 21 (2005), 19–24.
  • R. Ikehata and Y. Inoue, Total energy decay for semilinear wave equations with a critical potential type of damping, Nonlinear Anal., 69 (2008), 1396–1401.
  • R. Ikehata, G. Todorova and B. Yordanov, Critical exponent for semilinear wave equations with space-dependent potential, Funkcial. Ekvac., 52 (2009), 411–435.
  • T. Kato, Growth properties of solutions of the reduced wave equation with a variable coefficient, Comm. Pure Appl. Math., 12 (1959), 403–425.
  • A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci., 12 (1976-77), 169–189.
  • A. Matsumura, Energy decay of solutions of dissipative wave equations, Proc. Japan Acad. Ser. A Math. Sci., 53 (1977), 232–236.
  • A. Matsumura and N. Yamagata, Global weak solutions of the Navier-Stokes equations for multidimensional compressible flow subject to large external potential forces, Osaka J. Math., 38 (2001), 399–418.
  • N. Meyers, An expansion about infinity for solutions of linear elliptic equations, J. Math. Mech., 12 (1963), 247–264.
  • K. Mochizuki, Scattering theory for wave equations with dissipative terms, Publ. Res. Inst. Math. Sci., 12 (1976), 383–390.
  • K. Mochizuki and H. Nakazawa, Energy decay and asymptotic behavior of solutions to the wave equations with linear dissipation, Publ. Res. Inst. Math. Sci., 32 (1996), 401–414.
  • C. S. Morawetz, The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math., 14 (1961), 561–568.
  • M. Nakao, Energy decay for the linear and semilinear wave equations in exterior domains with some localized dissipations, Math. Z., 238 (2001), 781–797.
  • T. Narazaki, $L^{p}$-$L^{q}$ estimates for damped wave equations and their applications to semi-linear problem, J. Math. Soc. Japan, 56 (2004), 585–626.
  • K. Nishihara, Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping, J. Differential Equations, 137 (1997), 384–395.
  • K. Nishihara, $L^{p}$-$L^{q}$ estimates of solutions to the damped wave equation in $3$-dimensional space and their application, Math. Z., 244 (2003), 631–649.
  • K. Nishihara, Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term, Comm. Partial Differential Equations, 35 (2010), 1402–1418.
  • P. Radu, G. Todorova and B. Yordanov, Higher order energy decay rates for damped wave equations with variable coefficients, Discrete Conti. Dyn. Syst. Ser. S, 2 (2009), 609–629.
  • J. Rauch and M. Taylor, Decaying states of perturbed wave equations, J. Math. Anal. Appl., 54 (1976), 279–285.
  • M. Reissig, $L_{p}$-$L_{q}$ decay estimates for wave equations with time-dependent coefficients, J. Nonlinear Math. Phys., 11 (2004), 534–548.
  • W. A. Strauss, Nonlinear Wave Equations, CBMS Regional Conf. Ser. in Math., 73, Amer. Math. Soc., Providence, RI, 1989.
  • G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations, 174 (2001), 464–489.
  • G. Todorova and B. Yordanov, Weighted $L^2$-estimates of dissipative wave equations with variable coefficients, J. Differential Equations, 246 (2009), 4497–4518.
  • G. Todorova and B. Yordanov, Nonlinear dissipative wave equations with potential, Contemp. Math., 426 (2007), 317–337.
  • H. Uesaka, The total energy decay of solutions for the wave equation with a dissipative term, J. Math. Kyoto Univ., 20 (1980), 57–65.
  • J. Wirth, Solution representations for a wave equation with weak dissipation, Math. Methods Appl. Sci., 27 (2004), 101–124.
  • J. Wirth, Wave equations with time-dependent dissipation. I. Non-effective dissipation, J. Differential Equations, 222 (2006), 487–514.
  • J. Wirth, Wave equations with time-dependent dissipation. II. Effective dissipation, J. Differential Equations, 232 (2007), 74–103.