Evolutionary Pseudodifferential Equations with Smooth Symbols in S-Type Spaces.

We study an evolutionary equation with an operator (i∂/∂x), where φ is a smooth function satisfying certain conditions. As special cases of this equation, we get a partial differential equation of parabolic type with derivatives of finite and infinite orders and an equation with certain operators of fractional differentiation. It is shown that the restriction of the operator (i∂/∂x) to some spaces of type S coincides with a pseudodifferential operator constructed according to the function φ regarded as a symbol. We establish the correct solvability of a nonlocal multipoint (in time) problem for an equation of this kind with an initial function, which is an element of the space of generalized functions of ultradistribution type. The properties of the fundamental solution of this problem are analyzed. [ABSTRACT FROM AUTHOR]