Scattering theory for semilinear wave equations with small data in two space dimensions

Abstract

We study scattering theory for the semilinear wave equation Utt Au = JujP-lu in two space dimensions. We show that if p > p0 = (3+x/ii)/2, the scattering operator exists for smooth and small data. The lower bound p0 of p is considered to be optimal (see Glassey [6, 7], Schaeffer [ 18]). Our result is an extension of the results by Strauss [19], Klainerman [10], and Mochizuki and Motai [14, 15]. The construction of the scattering operator for small data does not follow directly from the proofs in [7, 13, 20 and 22] concerning the global existence of solutions for the Cauchy problem of the above equation with small initial data given at t = 0 in two space dimensions, because we have to consider the integral equation with unbounded integral region associated to the above equation: u(x, t) = u(x, t)? I[ [(II )(,s dy ds, ? 27r IJ Jix-yl<t-s /(tS)2 Ix y12 for t e R, where u-(x, t) is a solution of utt Au = 0 which u(x, t) approaches asymptotically as t -+ -oo. The proof of the basic estimate for the above integral equation is more difficult and complicated than that for the Cauchy problem of utt Au = Iu Pu in two space dimensions.