Mixed Neumann Problem Research Articles

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 e (z) is studied. The function f e (z) = Σ k ∈ K f k (z,e −1 Φ k (z),e), e∈[0,1], K is finite, f k (z,θ k ,e) are 2π-periodic with respect to θ k . The existence of solution u e on a domain z = (t,x,y) ∈ [0,T] x R + x R d , d = 1 or 2, is proved when e is sufficiently small ; T does not depend on e. By the non-linear geometric optics method the asymptotic (with respect to e → 0) solution u e is constructed. The estimation for the rest e 2 r e = u e − ũ e is derived and the limit r e , e → 0, is studied.