We consider $p$-evolution equations, for $p\geq2$, with complex valued coefficients. We prove that a necessary condition for $H^\infty$ 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 $|x|\to+\infty$.
"A Necessary condition for $H^\infty $ well-posedness of $p$-evolution equations." Adv. Differential Equations 21 (11/12) 1165 - 1196, November/December 2016.