Time-Nonlocal Problem for Evolutionary Equations with Pseudodifferential Operators in Spaces of Type S

We consider evolutionary equations with fractional differentiation operators whose restrictions to certain spaces of type S coincide with pseudodifferential operators constructed on the basis of smooth symbols, which are multipliers in these spaces. We establish the well-posedness of the time-nonlocal multipoint problem for these equations with initial function, which is an element of the space of generalized functions of the ultradistribution type. It is shown that the solutions of these problems stabilize to zero in the spaces of generalized functions of the type S′ (weak stabilization) and also stabilize to zero uniformly on \({\mathbb{R}}\) $\mathbb{R}$ in the case where the initial generalized function has a bounded support.