A Nonlocal in Time Problem for Evolutionary Singular Equations in Generalized Spaces of Type S∘

Hanna Verezhak,Vasyl Gorodetskyi

doi:10.1155/2020/6873414

Hanna Verezhak, Vasyl Gorodetskyi

Open Access

https://doi.org/10.1155/2020/6873414

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Abstract

In this paper, we establish the correct solvability of a nonlocal multipoint in time problem for the evolutionary equation of a parabolic type with the Bessel operator of infinite order in the case where the initial function is an element of the space of generalized functions of type S∘′.

Highlights

Singular parabolic equations with the Bessel operator are related to the equations with degeneracy in terms of the spatial variable operator, and they are close to evenly parabolic equations in terms of their internal properties
The Cauchy problem for singular parabolic equations in the classes of distributions and ultradistributions was studied by Ya.I
Functions from these spaces and all their derivatives decrease on the real axis as jxj ⟶ ∞ faster than exp f−a jxj1/αg, a > 0, x ∈ R: Such spaces are often used in the study of the problem of the classes of uniqueness and the classes of correct solvability of the Cauchy problem for partial differential equations

Summary

Introduction

Singular parabolic equations with the Bessel operator are related to the equations with degeneracy in terms of the spatial variable operator (such equations degenerate on the boundary of domain), and they are close to evenly parabolic equations in terms of their internal properties. 203–211) proposed a method of constructing functional spaces of infinitely differentiable functions given on R, which impose certain conditions for decreasing on infinity and increasing derivatives with increasing of order. These conditions are set using inequalities jxkφðnÞðxÞj ≤ ckn, fk, ng ⊂. Functions from these spaces and all their derivatives decrease on the real axis as jxj ⟶ ∞ faster than exp f−a jxj1/αg, a > 0, x ∈ R: Such spaces are often used in the study of the problem of the classes of uniqueness and the classes of correct solvability of the Cauchy problem for partial differential equations. The purpose of this work is to investigate a nonlocal multipoint time problem for evolutionary singular equations of infinite order in generalized spaces of type S∘

The Spaces of Test Functions

A Nonlocal Multipoint in Time Problem

Δt h T

Conclusion