Enhanced existence time of solutions to evolution equations of Whitham type

Abstract

We show that Whitham type equations u_t + u u_x -L u_x = 0, where L is a general Fourier multiplier operator of order α\in [-1,1], α\neq 0, allow for small solutions to be extended beyond their expected existence time. The result is valid for a range of quadratic dispersive equations with inhomogeneous symbols in the dispersive range given by α, and should be extendable to other equations of the same relative dispersive strength.

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