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Multidimensional Oscillations for Non‐linear Hyperbolic Mixed Problem: The Justified Non‐linear Geometric Optics Method
Author(s) -
Łada A.
Publication year - 1996
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(199602)19:3<217::aid-mma769>3.0.co;2-h
Subject(s) - mathematics , limit (mathematics) , mathematical analysis , function (biology) , domain (mathematical analysis) , mathematical physics , biology , evolutionary biology
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.

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