A necessary condition for $H^\infty$ well-posedness of p-evolution   equations
  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 goes to infinity. 
A. Ascanelli, C. Boiti, L. Zanghirati
math.AP