Decay and scattering of solutions of a nonlinear relativistic wave equation

Abstract. We consider the defocusing nonlinear wave equation u tt − Δu + |u| p u = 0 with spherically-symmetric initial data in the regime(which is energy-supercritical) and dimensions 3 ≤ d ≤ 6; we also consider d ≥ 7, but for a smaller range of p >. The principal result is that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that maximal-lifespan solutions with bounded critical Sobolev norm are global and scatter.

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“…[20,21]). It is a very close analogue of the formula used in [2] in the NLS setting; see also [18,29].…”

Section: Morawetz Inequalitymentioning

confidence: 99%

The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions

Killip

Vişan

2011

Proc. Amer. Math. Soc.

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“…It is valid only for certain classical one-dimensional nonlinear Klein-Gordon equations. Here M (t) represents the amplitude of a solution of (2.2) at time t, that is [26].…”

Section: Modified Klein-gordon Equationsmentioning

confidence: 99%

A numerical method for computing radially symmetric solutions of a dissipative nonlinear modified Klein‐Gordon equation

Macías‐Díaz

Puri

2005

Numerical Methods Partial

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In this paper we develop a finite-difference scheme to approximate radially symmetric solutions of the initial-value problem with smooth initial conditions (1)in an open sphere D around the origin, where the internal and external damping coefficients-β and γ, respectively-are constant, and the nonlinear term has the form G ′ (w) = w p , with p > 1 an odd number. The functions φ and ψ are radially symmetric in D, and φ, ψ, rφ and rψ are assumed to be small at infinity. We prove that our scheme is consistent order O(∆t 2 ) + O(∆r 2 ) for G ′ identically equal to zero, and provide a necessary condition for it to be stable order n. Part of our study will be devoted to compare the physical effects of β and γ.

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“…There are amount of papers concerning the asymptotic behavior of solutions for the nonlinear Schrö dinger equation (see [3,6,9,10,11,12,14,17,18,22,24,25,26,27,35,36,37,38,39,40,41]), and for the nonlinear Klein-Gordon equation (see [4,5,13,20,19,21,23,24,28,30,34,35,36,37]). We consider the existence of wave operators W G .…”

Section: > < > : ðKgsþmentioning

confidence: 99%

Wave Operators for the Coupled Klein-Gordon-Schrödinger Equations in Two Space Dimensions

Shimomura

2004

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Abstract. In this paper, the scattering theory for the coupled Klein-GordonSchrö dinger equation with the Yukawa type interaction in two space dimensions is studied. The scattering problem for this equation belongs to the borderline between the short range case and the long range one. The existence of the wave operators to this equation for small scattered states is proved without any restrictions on the support of the Fourier transform of them.

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Decay and scattering of solutions of a nonlinear relativistic wave equation

Cited by 180 publications

References 12 publications

The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions

The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions

A numerical method for computing radially symmetric solutions of a dissipative nonlinear modified Klein‐Gordon equation

Wave Operators for the Coupled Klein-Gordon-Schrödinger Equations in Two Space Dimensions

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