Remarks on Keel-Smith-Sogge estimates and some applications to nonlinear higher-order wave equations. (Q417001)

The author gives a generalization of the Keel-Smith-Sogge estimate (see \textit{M. Keel, H. F. Smith} and \textit{Ch. D. Sogge} [J. Anal. Math. 87, 265--279 (2002; Zbl 1031.35107)]) for the higher-order wave equation \((u_{tt}+(-\Delta )^m)u=F\), \(t\geq 0\), \(x\in \mathbb {R}^n\) (\(n\geq 1\), \(m>0\)). He applies the results to the Cauchy problem with \(F\) being a function of \(u\) and its derivatives showing the existence of a global and almost global solutions.