Remark on the asymptotic behavior of the klein gordon equation in ℝn+1

✍ Scribed by S. Klainerman

• Publisher: John Wiley and Sons
• Year: 1993
• Tongue: English
• Weight: 275 KB
• Volume: 46
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/cpa.3160460202

✦ Synopsis

Dedicated to the 80-th anniversary of F. John.

Consider the Klein-Gordon equation in Minkowski space-time Rflfi. Here 0 = f"fia,aB denotes the D'Alemberton operators with f the Minkowski metric of W I . Relative to inertial coordinates x n , ct = 0, 1 ,..., n, we have fob = diag(-1, l , . . ., 1).

As usual we write xo = t and x = (x', . . . , x"). Throughout the paper we use the standard conventions summation and raising and lowering of indices with respect to the metric fap.

In what follows we plan to reexamine the asymptotic behavior of solutions to (1 ) subject to initial conditions, on t = 0. We recall that in [l], [2] the asymptotic behavior of ( l ) , (2) was derived using only energy estimates and the commutation properties of ( 1 ) with respect to the generators of the Lorentz group (3) These types of estimates are crucial in the study of nonlinear perturbations of (1) as shown in [l]. Recently in [3] T. Sideris derived estimates for the inhomogeneous K-G equation using once more the vector fields (3). His estimates, which can also be applied to general nonlinear perturbations of ( l ) , are based on a careful analysis of the fundamental solution of the Klein-Gordon operator and do not require the assumption of compact support for f , g as in [I], [2]. Nevertheless the estimates of [3] as well as those of [l],

[2] do not do complete justice to the character of the asymptotic behavior of solutions to (1). Indeed, one of the main feature of this behavior, which can be easily derived by the method of stationary phase (see [2]) or directly from the explicit form of the solutions, is that the decay in time of 4 is much faster along null directions than along time-like directions. This fact