Enhanced existence time of solutions to evolution equations of Whitham type

Abstract

<p style='text-indent:20px;'>We show that Whitham type equations <inline-formula><tex-math id="M1">\begin{document}$u_t + u u_x -\mathcal{L} u_x = 0$\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$L$\end{document}</tex-math></inline-formula> is a general Fourier multiplier operator of order <inline-formula><tex-math id="M3">\begin{document}$\alpha \in [-1, 1]$\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$\alpha\neq 0$\end{document}</tex-math></inline-formula>, allow for small solutions to be extended beyond their ordinary existence time. The result is valid for a range of quadratic dispersive equations with inhomogenous symbols in the dispersive regime given by the parameter <inline-formula><tex-math id="M5">\begin{document}$\alpha$\end{document}</tex-math></inline-formula>.</p>