Space-time fractional Cauchy problem in spaces of generalized functions

• Date: 2012
• Main Authors: Lopushanskaya, G. P., Lopushanskyi, A. O., Лопушанська, Г. П., Лопушанський, А. О.
• Format: Article
• Language: Ukrainian; English
• Published: Institute of Mathematics, NAS of Ukraine 2012
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/2641
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

We prove a theorem on the existence and uniqueness and obtain a representation using the Green vector function for the solution of the Cauchy problem $$u^{(\beta)}_t + a^2(-\Delta)^{\alpha/2}u = F(x, t), \quad (x, t) \in \mathbb{R} ^n \times (0, T], \quad a = \text{const} $$ $$u(x, 0) = u_0(x), \qua...