On small data scattering for 2-dimensional semilinear wave equations. (English) Zbl 0768.35063

Kubota, Kôji; Mochizuki, Kiyoshi

Hokkaido Math. J. 22, No. 1, 79-97 (1993).

The authors consider the wave equation in two dimensions, with nonlinear zero-order term of the type \(| u|^{\rho-1}u\) or \(| u|^ \rho\), where \(\rho\) is greater than some critical value. Comparing it with the free linear wave equation, they show the existence of the scattering operator acting on a dense set of a neighborhood of zero in the energy norm. More precisely, for sufficiently small and regular initial data, they construct a solution of the nonlinear equation behaving at \(-\infty\) as the solution of the linear initial-value problem, and whose behaviour at \(+\infty\) is the same as some other solution of the free wave equation.

Reviewer: A.Martinez (Villetaneuse)

MSC:

35P25

Scattering theory for PDEs

Keywords:

nonlinear wave equation; nonlinear scattering