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Zbl 0596.35109
Albert, John
Dispersion of low-energy waves for the generalized Benjamin-Bona-Mahony equation. (English)
[J] J. Differ. Equations 63, 117-134 (1986). ISSN 0022-0396

The generalizations of the equations in the title $$ (1)\quad u\sb t+u\sb x+(F(u))\sb x+u\sb{xxx}=0\quad and\quad (2)\quad u\sb t+u\sb x+(F(u))\sb x-u\sb{xxt}=0 $$ are considered. The main result is the following analogue of a result of Strauss for the equation (1). Theorem: Let $F: R\to R$ be a $C\sp{\infty}$ function such that $\vert F'(s)\vert =O(\vert s\vert\sp{6+\epsilon})$ as $s\to 0$ for some $\epsilon >0$. Then there exists a number $\delta\sb F>0$ such that if I.V.P. for (2) at the initial function $u(x,0)=\phi (x)$ for which $\phi \in C\sp 2\sb b\cap N$ and $\vert \phi \vert\sb N<\delta\sb F$, then the solution satisfies $\vert u(x,t)\vert \le A(1+t)\sp{-1/3}$ for all $x\in R$ and $t\ge 0$, where A is independent of x and t.
[V.Kostova]

MSC 2000:: *35Q99 PDE of mathematical physics and other areas
81U30 Dispersion theory, dispersion relations (quantum theory)

Keywords: Benjamin-Bona-Mahony equation; Korteweg-de Vries equation; solitary-wave solutions

Cited in: Zbl 0697.35116

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