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Dispersion of low-energy waves for the generalized Benjamin-Bona-Mahony equation. (English) Zbl 0596.35109

The generalizations of the equations in the title

(1)u t +u x +(F(u)) x +u xxx =0and(2)u t +u x +(F(u)) x -u xxt =0

are considered. The main result is the following analogue of a result of Strauss for the equation (1). Theorem: Let F:RR be a C function such that |F ' (s)|=O(|s| 6+ϵ ) as s0 for some ϵ>0. Then there exists a number δ F >0 such that if I.V.P. for (2) at the initial function u(x,0)=ϕ(x) for which ϕC b 2 N and |ϕ| N <δ F , then the solution satisfies |u(x,t)|A(1+t) -1/3 for all xR and t0, where A is independent of x and t.

Reviewer: V.Kostova

MSC:
35Q99PDE of mathematical physics and other areas
81U30Dispersion theory, dispersion relations (quantum theory)