Abstract
Spaces of $S$ type, introduced by I.Gelfand and G.Shilov, as well as spaces of type $S'$, topologically conjugate with them, are natural sets of the initial data of the Cauchy problem for broad classes of equations with partial derivatives of finite and infinite orders, in which the solutions are integer functions over spatial variables. Functions from spaces of $S$ type on the real axis together with all their derivatives at $|x|\to \infty$ decrease faster than $\exp\{-a|x|^{1/\alpha}\}$, $\alpha > 0$, $a > 0$, $x\in \mathbb{R}$. The paper investigates a nonlocal multipoint by time problem for equations with partial derivatives of parabolic type in the case when the initial condition is given in a certain space of generalized functions of the ultradistribution type ($S'$ type). Moreover, results close to the Cauchy problem known in theory for such equations with an initial condition in the corresponding spaces of generalized functions of $S'$ type were obtained. The properties of the fundamental solution of a nonlocal multipoint by time problem are investigated, the correct solvability of the problem is proved, the image of the solution in the form of a convolution of the fundamental solution with the initial generalized function, which is an element of the space of generalized functions of $S'$ type.
Highlights
Âñòàíîâëåíî êîðåêòíó ðîçâ'ÿçíiñòü íåëîêàëüíîáàãàòîòî÷êîâîçà ÷àñîì çàäà÷i äëÿ îäíîãî êëàñó ðiâíÿíü ïàðàáîëi÷íîãî òèïó ç ïî÷àòêîâîþ óìîâîþ, ÿêà çàäà1òüñÿ â ïðîñòîði óçàãàëüíåíèõ ôóíêöié òèïó óëüòðàðîçïîäiëiâ.
Êëþ÷îâi ñëîâà i ôðàçè: íåëîêàëüíà áàãàòîòî÷êîâà çà ÷àñîì çàäà÷à, ïðîñòið óçàãàëüíåíèõ ôóíêöié, ôóíäàìåíòàëüíèé ðîçâ'ÿçîê, ðiâíÿííÿ ïàðàáîëi÷íîãî òèïó.
Óçàãàëüíåííÿì çàäà÷i Êîøi äëÿ òàêèõ ðiâíÿíü 1 íåëîêàëüíà áàãàòîòî÷êîâà çà ÷àñîì
Summary
Âñòàíîâëåíî êîðåêòíó ðîçâ'ÿçíiñòü íåëîêàëüíîáàãàòîòî÷êîâîçà ÷àñîì çàäà÷i äëÿ îäíîãî êëàñó ðiâíÿíü ïàðàáîëi÷íîãî òèïó ç ïî÷àòêîâîþ óìîâîþ, ÿêà çàäà1òüñÿ â ïðîñòîði óçàãàëüíåíèõ ôóíêöié òèïó óëüòðàðîçïîäiëiâ. Êëþ÷îâi ñëîâà i ôðàçè: íåëîêàëüíà áàãàòîòî÷êîâà çà ÷àñîì çàäà÷à, ïðîñòið óçàãàëüíåíèõ ôóíêöié, ôóíäàìåíòàëüíèé ðîçâ'ÿçîê, ðiâíÿííÿ ïàðàáîëi÷íîãî òèïó. Óçàãàëüíåííÿì çàäà÷i Êîøi äëÿ òàêèõ ðiâíÿíü 1 íåëîêàëüíà áàãàòîòî÷êîâà çà ÷àñîì S S íÿíü ç ÷àñòèííèìè ïîõiäíèìè ïàðàáîëi÷íîãî òèïó â ïðîñòîðàõ òèïó òà ′, ïðè öüîìó îäåðæàíî ðåçóëüòàòè, áëèçüêi äî âiäîìèõ ó òåîðiçàäà÷i Êîøi äëÿ òàêèõ ðiâíÿíü ç ïî÷àòêîâèìè óìîâàìè â ïðîñòîðàõ óçàãàëüíåíèõ ôóíêöié òèïó óëüòðàðîçïîäiëiâ [4, 5].
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