Evolutionary pseudodifferential equations with smooth symbols in the $S$-type spaces

• Дата: 2023
• Автори: Horodets’kyi, V., Petryshyn, R., Martynyuk, O., Городецький, Василь, Петришин, Роман, Мартинюк, Ольга, Мартинюк, Ольга Василівна
• Формат: Стаття
• Мова: Українська
• Опубліковано: Institute of Mathematics, NAS of Ukraine 2023
• Онлайн доступ: https://umj.imath.kiev.ua/index.php/umj/article/view/7443
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

UDC 517.98 We study an evolutionary equation with an operator $\varphi(i \partial /\partial x),$ where $\varphi$ is a smooth function  satisfying certain conditions.  As a special case of this equation, we get a partial differential equation of parabolic type with derivatives of finite and infinite orders and a certain equation with operators of fractional differentiation. It is established that the restriction of the operator $\varphi(i \partial / \partial x)$ to some spaces of type $S$ coincides with a pseudodifferential operator constructed according to the function $\varphi$ 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 to this problem are analyzed.