On One Evolution Equation of Parabolic Type with Fractional Differentiation Operator in S Spaces

Abstract

In the paper, we investigate a nonlocal multipoint by a time problem for the evolution equation with the operator $A={\left(I-\mathrm{\Delta }\right)}^{w/2}$, $\mathrm{\Delta }=\left({\text{d}}^2/\text{d}{x}^2\right),$and $\omega \in [1;-2)$ is a fixed parameter. The operator $A$ is treated as a pseudodifferential operator in a certain space of type $S$. The solvability of this problem is proved. The representation of the solution is given in the form of a convolution of the fundamental solution with the initial function which is an element of the space of generalized functions of ultradistribution type. The properties of the fundamental solution are investigated. The behavior of the solution at $t\to +\infty$ (solution stabilization) in the spaces of generalized functions of type $S\text{'}$ and the uniform stabilization of the solution to zero on $ℝ$ are studied.