# Differences between parabolic operators of second order and higher order

Properties of parabolic operators of second order have been extensively studied, such as the existence or uniqueness theorem. In higher order case ($u_t-P(D)u$, where $P$ is a $2m$ order uniformly elliptic partial differential operators), similar results are also valid. But one of the basic differences in higher order case is the lack of maximal principle(or the fundamental solution is not positive), also the Feymann-Kac formular for the solution of heat equation $u_t-\Delta u+Vu=0$ has no analogs in higher order case.

I want to know some other other differences(such as the regularity property) between the higher order and second order case, and the reason behind it, so I can get a more clear picture of the higher order parabolic operators.

At last, I want to know if there are some papers or books dealing with such topics. BTW, the only book I know that related to higher order parabolic operators is A.Friedman's partial differential equations of parabolic type.

Thanks in advance for any comment or reference.

• There is a book by Eidelman on parabolic systems. – timur Nov 25 '13 at 4:11