Space-time fractional Cauchy problem in spaces of generalized functions

  • G. P. Lopushanskaya (Львiв. нац. ун-т, Ряшiв. ун-т, Польща)
  • A. O. Lopushanskyi (Львiв. нац. ун-т, Ряшiв. ун-т, Польща)

Abstract

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 $\mathfrak{F}[(-\Delta)^{\alpha/2} \psi(x)] = |\lambda|^{\alpha} \mathfrak{F}[\psi(x)]$.
Published
25.08.2012
How to Cite
LopushanskayaG. P., and LopushanskyiA. O. “Space-Time Fractional Cauchy Problem in Spaces of Generalized Functions”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, no. 8, Aug. 2012, pp. 1067-79, http://umj.imath.kiev.ua/index.php/umj/article/view/2641.
Section
Research articles