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We consider the Cauchy problem for systems of nonlinear wave equations withmultiple propagation speeds in three space dimensions. Under the null conditionfor such systems, the global existence of small amplitude solutions is known.In this paper, we will show that the global solution is asymptotically free inthe energy sense, by obtaining the asymptotic pointwise behavior of thederivatives of the solution. Nonetheless we can also show that the pointwisebehavior of the solution itself may be quite different from that of the freesolution. In connection with the above results, a theorem is also developed tocharacterize asymptotically free solutions for wave equations in arbitraryspace dimensions.

Asymptotic bahavior for systems of nonlinear wave equations with multiple propagation speeds in three space dimensions