Multidimensional Oscillations for Non‐linear Hyperbolic Mixed Problem: The Justified Non‐linear Geometric Optics Method

The mixed‐Neumann problem for the non‐linear wave equation □ u − a ( u )(∣∂ t u )∣ 2 −∣∇ u ∣ 2 = f ε ( z ) is studied. The function f ε ( z ) = ∑ k ∈ K f k ( z ,ε −1 ϕ k ( z ),ε), ε∈[0,1], K is finite, f k ( z ,θ k ,ε) are 2π‐periodic with respect to θ k . The existence of solution u ε on a domain z = ( t,x,y )∈[0, T ]×ℝ + ×ℝ d , d = 1 or 2, is proved when ε is sufficiently small; T does not depend on ε. By the non‐linear geometric optics method the asymptotic (with respect to ε→0) solution ũ ε is constructed. The estimation for the rest ε 2 r ε = u ε −ũ ε is derived and the limit r ε , ε→0, is studied.