2017
Том 69
№ 5

All Issues

Space-time fractional Cauchy problem in spaces of generalized functions

Lopushanskaya G. P., Lopushanskyi A. O.

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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 $\mathfrak{F}[(-\Delta)^{\alpha/2} \psi(x)] = |\lambda|^{\alpha} \mathfrak{F}[\psi(x)]$.

English version (Springer): Ukrainian Mathematical Journal 64 (2012), no. 8, pp 1215-1230.

Citation Example: Lopushanskaya G. P., Lopushanskyi A. O. Space-time fractional Cauchy problem in spaces of generalized functions // Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1067-1079.

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