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We study the Cauchy problem for a system of semi-linear coupledfractional-diffusion equations with polynomial nonlinearities posed in $%\mathbb{R}_{+}\times \mathbb{R}^{N}$. Under appropriate conditions on theexponents and the orders of the fractional time derivatives, we present acritical value of the dimension N, for which global solutions with small dataexist, otherwise solutions blow-up in finite time. Furthermore, the large timebehavior of global solutions is discussed.

Existence of global solutions and blow-up of solutions for coupled systems of fractional diffusion equations