Space–time fractional Cauchy problem in spaces of generalized functions

We prove the theorem on existence and uniqueness of solution to the Cauchy problem

$$ \begin{array}{*{20}{c}} {u_t^{{\left( \beta \right)}}+{a^2}{{{\left( {-\varDelta } \right)}}^{{\alpha /2}}}u=F\left( {x,t} \right),\quad \left( {x,t} \right)\in {{\mathbb{R}}^n}\times (0,T],\quad a=\mathrm{const}\hbox{,}} \\ {u\left( {x,0} \right)={u_0}(x),\quad x\in {R^n},} \\ \end{array} $$

where $u_t^{{(\beta )}}$ is the Riemann–Liouville fractional derivative of order β ∈ (0, 1) and u ₀ and F belong to spaces of generalized functions. A representation of this solution is obtained by using the vector Green function. We also establish the character of singularities of the solution for t = 0 depending on the order of singularity of a given generalized function in the initial condition and the character of power singularities of the function on the right-hand side of the equation. In this case, the fractional n-dimensional Laplace operator is defined by using the Fourier transformation $\mathfrak{F}\left[ {{{{\left( {-\varDelta } \right)}}^{{{\alpha \left/ {2} \right.}}}}\psi (x)} \right]={{\left| \lambda \right|}^{\alpha }}\mathfrak{F}\left[ {\psi (x)} \right]$.