Decay rates for cubic and higher order nonlinear wave equations on asymptotically flat spacetimes

. In this paper, we prove pointwise decay rates for cubic and higher order nonlinear wave equations, including quasilinear wave equations, on asymptotically flat and time-dependent spacetimes. We assume that the solution to the linear equation (rather than the nonlinear equation) satisfies a weaker form of the standard integrated local energy decay, or Morawetz, estimate. For nonlinearities with a total derivative structure, we prove better pointwise decay rates.

[1]  Shi-Zhuo Looi.  Pointwise decay for the energy-critical nonlinear wave equation , 2022, 2205.13197.

[2]  M. Tohaneanu,et al.  Global existence and pointwise decay for the null condition , 2022, 2204.03626.

[3]  Shi-Zhuo Looi.  Pointwise decay for the wave equation on nonstationary spacetimes , 2021, Journal of Mathematical Analysis and Applications.

[4]  Mihai Tohaneanu,et al.  Scattering for critical wave equations with variable coefficients , 2019, Proceedings of the Edinburgh Mathematical Society.

[5]  D. Tataru,et al.  Local energy decay for scalar fields on time dependent non-trapping backgrounds , 2017, American Journal of Mathematics.

[6]  I. Rodnianski,et al.  Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case |a| < M , 2014, 1402.7034.

[7]  M. Tohaneanu,et al.  Price's Law on Nonstationary Space-Times , 2011, 1104.5437.

[8]  N. Szpak,et al.  Global Pointwise Decay Estimates for Defocusing Radial Nonlinear Wave Equations , 2009, 0903.0799.

[9]  J. Bony,et al.  The Semilinear Wave Equation on Asymptotically Euclidean Manifolds , 2008, 0810.0464.

[10]  N. Szpak.  Linear and nonlinear tails I: general results and perturbation theory , 2007, 0710.1782.

[11]  中村 誠.  Global existence of solutions to multiple speed systems of quasilinear wave equations in exterior domains (非線型波動及び分散型方程式に関する研究 短期共同研究報告集) , 2005 .

[12]  I. Rodnianski,et al.  The global stability of the Minkowski space-time in harmonic gauge , 2004, math/0411109.

[13]  I. Rodnianski,et al.  Global Existence for the Einstein Vacuum Equations in Wave Coordinates , 2003, math/0312479.

[14]  C. Sogge,et al.  Hyperbolic trapped rays and global existence of quasilinear wave equations , 2003, math/0301290.

[15]  Hart F. Smith,et al.  Almost global existence for some semilinear wave equations , 2001, math/0108016.

[16]  J. Shatah,et al.  Decay estimates for the critical semilinear wave equation , 1998 .

[17]  Thomas C. Sideris,et al.  On almost global existence for nonrelativistic wave equations in 3D , 1996 .

[18]  M. Grillakis.  Regularity and asymptotic behavior of the wave equation with a critical nonlinearity , 1990 .

[19]  Darryl D. Holm,et al.  Nonlinear systems of partial differential equations in applied mathematics , 1986 .

[20]  Demetrios Christodoulou,et al.  Global solutions of nonlinear hyperbolic equations for small initial data , 1986 .

[21]  Jalal Shatah,et al.  Global existence of small solutions to nonlinear evolution equations , 1982 .

[22]  H. Pecher.  Decay of solutions of nonlinear wave equations in three space dimensions , 1982 .

[23]  W. Wahl.  Lp-Decay rates for homogeneous wave-equations , 1971 .

[24]  J. Ralston.  Solutions of the wave equation with localized energy , 1969 .

[25]  Cathleen S. Morawetz,et al.  Time decay for the nonlinear Klein-Gordon equation , 1968, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[26]  Long Time Behaviour of Solutions to Nonlinear Wave Equations , 2010 .

[27]  Luis Vega,et al.  On the Zakharov and Zakharov-Schulman Systems , 1995 .

[28]  Gustavo Ponce,et al.  Global, small amplitude solutions to nonlinear evolution equations , 1983 .

[29]  Sergiu Klainerman,et al.  Global existence for nonlinear wave equations , 1980 .