TY - JOUR
AU - G. P. Lopushanskaya
AU - A. O. Lopushanskyi
PY - 2012/08/25
Y2 - 2024/04/18
TI - Space-time fractional Cauchy problem in spaces of generalized functions
JF - Ukrainsâ€™kyi Matematychnyi Zhurnal
JA - Ukr. Mat. Zhurn.
VL - 64
IS - 8
SE - Research articles
DO -
UR - https://umj.imath.kiev.ua/index.php/umj/article/view/2641
AB - 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$$u^{(\beta)}_t + a^2(-\Delta)^{\alpha/2}u = F(x, t), \quad (x, t) \in \mathbb{R} ^n \times (0, T], \quad a = \text{const} $$$$u(x, 0) = u_0(x), \quad x \in \mathbb{R} ^n$$where $u^{(\beta)}_t$ is the Riemann-Liouville fractional derivative of order $\beta \in (0,1)$, and $u_0$ and $F$ belong to some spaces of generalized functions. We also establish the character of the singularity of the solution at $t = 0$ and its dependence on the order of singularity of the given generalized function in the initial condition and the character of the power singularities of the function on right-hand side of the equation. Here, the fractional $n$-dimensional Laplace operator$\mathfrak{F}[(-\Delta)^{\alpha/2} \psi(x)] = |\lambda|^{\alpha} \mathfrak{F}[\psi(x)]$.
ER -