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Wave operators to a quadratic nonlinear Klein-Gordon equation in two space dimensions. (English) Zbl 1173.35599

Summary: We study asymptotics around the final states of solutions to the nonlinear Klein-Gordon equations with quadratic nonlinearities in two space dimensions ( t 2 -Δ+m 2 )u=λu 2 , (t,x)× 2 , where λ. We prove that if the final states

u 1 + q q-1 4-4 q ( 2 ) 5 2,1 ( 2 ) 1 2 ( 2 ),u 2 + q q-1 3-4 q ( 2 ) 3 2,1 ( 2 ) 1 1 ( 2 ),

and u 1 + 1 2 +u 2 + 1 1 is sufficiently small, where 4<q, then there exists a unique global solution u([T,);𝕃 2 ( 2 )) to the nonlinear Klein-Gordon equations such that u(t) tends as t in the 𝕃 2 sense to the solution u 0 (t)=u 1 + cos((i) m t)+((i) m -1 u 2 + )sin((i) m t) of the free Klein-Gordon equation.

MSC:
35L70Nonlinear second-order hyperbolic equations
35B40Asymptotic behavior of solutions of PDE