A necessary condition for \(H^\infty \) well-posedness of \(p\)-evolution equations. (English) Zbl 1375.35090

Ascanelli, Alessia; Boiti, Chiara; Zanghirati, Luisa

Adv. Differ. Equ. 21, No. 11-12, 1165-1196 (2016).

Summary: 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 \(|x|\to+\infty\).

MSC:

35G10	Initial value problems for linear higher-order PDEs
35A27	Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs