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On the decay of solutions of the generalized Benjamin-Bona-Mahony equation. (English) Zbl 0697.35116
It is shown that, for all integers \(p>4\), solutions of the generalized Benjamin-Bona-Mahony equation \[ u_ t+u_ x+u^ pu_ x-u_{xxt}=0 \] satisfy the decay estimate \(\sup_{0<t<\infty,-\infty <x<\infty}(1+t)^{1/3}| u(x,t)| <\infty\); provided the initial data u(x,0) is sufficiently small in the norm of \(W^{1,2}({\mathbb{R}})\cap W^{2,5}({\mathbb{R}}).\) (The proof depends on the estimate \[ | v(\cdot,t)|_{L^{\infty}}\leq c(| v(\cdot,t)|_{L^ 1}+| v(\cdot,t)|_{W^{2,4}})(1+t)^{-1/3} \] for solutions of the linear equation \(v_ t+v_ x-v_{xxt}=0.)\)
This improves earlier results of the authors for \(p>6\) [J. Differ. Equations 63, 117-134 (1986; Zbl 0596.35109)]. Analogous results for the generalized Korteweg-de Vries equation have been proved by W. A. Strauss [Arch. Ration. Mech. Anal. 55, 86-92 (1974; Zbl 0289.35048)]; by G. Ponce and L. Vega [to appear in J. Funct. Anal.]; and by M. Christ and M. Weinstein [preprint].
Reviewer: J.Albert

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
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