Existence of mild solutions for a system of partial differential equations with time-dependent generators

We give sufficient conditions for global existence of positive mild solutions for the weak coupled system: \begin{eqnarray*} \frac{\partial u_{1}}{\partial t} &=&ρ_{1}t^{ρ_{1}-1}Δ_{α_{1}}u_{1}+t^{σ_{1}}u_{2}^{β_{1}},\ \ u_{1}\left(0\right) =φ_{1}, \\ \frac{\partial u_{2}}{\partial t} &=&ρ_{2}t^{ρ_{2}-1}Δ_{α_{2}}u_{2}+t^{σ_{2}}u_{1}^{β_{2}},\ \ u_{2}\left(0\right) =φ_{2}, \end{eqnarray*} where $Δ_{α_{i}}$ is a fractional Laplacian, $0<α_{i}\leq 2,\ β_{i}>1,\ ρ_{i}>0,\ σ_{i}>-1\$are constants and the initial data $φ_{i}$ are positive, bounded and integrable functions.

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