This paper is concerned with the initial value problem for semilinear systems of wave equations. First we show a global existence result for small amplitude solutions to the systems. Then we study asymptotic behavior of the global solution. We underline that "modified" free profiles are obtained for all global solutions to the systems even in the case where the free profile might not exist. Moreover, we prove non-existence of any free profiles for the global solution in some cases where the effect of the nonlinearity is strong enough.
We consider a nonlinear system of two-dimensional Klein-Gordon equations with masses m 1 , m 2 satisfying the resonance relation m 2 = 2m 1 > 0. We introduce a structural condition on the nonlinearities under which the solution exists globally in time and decays at the rate O(|t| −1 ) as t → ±∞ in L ∞ . In particular, our new condition includes the Yukawa type interaction, which has been excluded from the null condition in the sense of J.-M.Delort, D.Fang and R.Xue (J.Funct. Anal.211(2004), 288-323).
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