A necessary condition for $H^{\mathrm{\infty }}$ well-posedness of p-evolution equations

We consider p-evolution equations, for $p\ge 2$, with complex valued coefficients. We prove that a necessary condition for $H^{\mathrm{\infty }}$ well-posedness of the associated Cauchy problem is that the imaginary part of the coefficient of the subprincipal part (in the sense of Petrowski) satisfies a decay estimate as x goes to infinity.