95-322 Brenner A.
THE MIXED PROBLEMS FOR MULTIDIMENSIONAL TIME POLYPARABOLIC OPERATORS (261K, LaTeX) Jun 28, 95
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Abstract. We introduce a new class of polyparabolic equations with multidimensional time and develop the existence theory for the respective nonhomogeneous mixed problems. Also, we obtain the uniqueness theorem for the problems satisfying the newly defined inductive polyparabolicity condition. Various equations of this type occur in the theory of probability, cosmology and physics. In particular, they include the class of the so-called ultraparabolic equations. In order to obtain the existence and uniqueness for polyparabolic problems we study elliptic boundary value problems with parameters in a bounded domain. For example, the notion of the ellipticity with parameters for differential and pseudodifferential operator pencils can be applied to the resolvent construction which leads to the definition of the corresponding complex powers and $\ \zeta -$function, i.e. to a new functional calculus of operators with promising applications to the multiparameter spectral theory.

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