On fractional parabolic systems of vector order

The paper considers the Cauchy problem for the system of partial differential equations of fractional order DₜB U(t,x) + A(D) U (t,x)=H(t,x). Here U and H are vector-functions, the m× m matrix of differential operators 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 B has different components βⱼ∈ (0,1], and βⱼ 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: βⱼ=β, $j=1,...,m$.