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

✍ Scribed by A. Łada

• Publisher: John Wiley and Sons
• Year: 1996
• Tongue: English
• Weight: 669 KB
• Volume: 19
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/(sici)1099-1476(199602)19:3<217::aid-mma769>3.0.co;2-h

✦ Synopsis

Communicated by A. Piskorek The mixed-Neumann problem for the non-linear wave equation m ua(u)(la,u)12 -IVu12) =f.(z) is studied. The function f , ( z ) = ~, , , &( z , E -~& ( z ) , E ) , E E [0, 11, K is finite, &(z,&,E) are 2s-periodic with respect to 0,. The existence of solution u, on a domain z = (t, x, y ) E [0, T ] x R + x Rd, 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 ti, is constructed. The estimation for the rest &*re = u,r?, is derived and the limit re, E + 0, is studied.