Brenner A.
THE MIXED PROBLEMS
FOR MULTIDIMENSIONAL TIME POLYPARABOLIC OPERATORS
(261K, LaTeX)
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.