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.

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

Any uniformly elliptic differential operator of even order (with the right sign!) generates an analytic semigroup, so there will not be much difference when it comes to regularity of solutions of a parabolic equation.

As you suspect, the real difference is in the theory of qualitative properties. Essentially, the whole theory of Dirichlet forms is not applicable to the case of higher orders, so the whole interplay with stochastic analysis is lost - this is not really surprising, admittedly. I am not aware of any kernel estimates for the semigroup generated by higher order differential operators, by the way.

  • $\begingroup$ Thanks for the answer, about the heat kernel, I think E.B. Davies have done some important work on this area, e.g. see his 95 paper "uniformly elliptic operator with measurable coefficient" or Barbatis's "sharp heat kernel estimates for higher order estimates with singular coefficients" and references therein. $\endgroup$
    – Tomas
    Nov 25 '13 at 14:11
  • $\begingroup$ Thanks! Very interesting papers, I was not aware of them. $\endgroup$ Nov 25 '13 at 14:27
  • $\begingroup$ Can you please elaborate a bit on your comment about the Dirichlet forms? Are you referring to the stuff related to coercivity, Gårding inequality etc? $\endgroup$
    – timur
    Dec 20 '13 at 21:17
  • $\begingroup$ No, I am referring to everything has to do with the Beurling-Deny conditions: The form associated with a bi-Laplacian with Dirichlet boundary conditions is still coercive, but e.g. it generates a semigroup that is not positive -- i.e., no parabolic maximum principle. $\endgroup$ Dec 21 '13 at 11:50

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