Abstract

We study the ill-posedness of a quasilinear wave equation. It was shown by Smith and Tataru in 2005 that for any $s>\frac{7}{4}$ (or $\frac{11}{4}$ in our situation), the equation is well-posed in $H^s\times H^{s-1}$. We show a sharpness result by exhibiting a quasilinear wave equation and an initial data such that the Cauchy problem is ill-posed for in $H^{11∕4}{(\ln <!--FUNCTION\; APPLICATION-->H)}^{-\beta }\times H^{7∕4}{(\ln <!--FUNCTION\; APPLICATION-->H)}^{-\beta }$.

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