On fractional parabolic systems of vector order

Abstract

The paper considers the Cauchy problem for the system of partial differential equations of fractional order $D_t^{\mathcal{B}}U(t,x)+\mathbb{A}(D)U(t,x)=H(t,x)$. Here $U$ and $H$ are vector-functions, the $m\times m$ matrix of differential operators $\mathbb{A}(D)$ is triangular (elements above or below the diagonal are zero). Operators located on the diagonal are elliptic. The main distinctive feature of this system is that the vector-order $\mathcal{B}$ has different components ${\beta }_j\in (0,1]$, and ${\beta }_j$ are not necessarily rational. Sufficient conditions (in some cases they are necessary) on the initial function and the right-hand side of the equation are found to ensure the existence of a classical solution. Note that the existence of a classical solution to systems of fractional differential equations was studied by various authors, but in all these works the fractional order had the same components for each equation: ${\beta }_j=\beta$, $j=1,...,m$.