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We investigate nonnegative solutions $u(x,t)$ and $v(x,t)$ of the nonlinearsystem of inequalities \[0\leq(\partial_t -\Delta)^\alpha u\leq v^\lambda\] \[0\leq (\partial_t -\Delta)^\beta v\leq u^\sigma\] in $\mathbb{R}^n\times\mathbb{R}$, $n\geq 1$, satisfying the initial conditions \[  u=v=0\quad\text{ in }\mathbb{R}^n \times(-\infty,0) \] where$\lambda,\sigma,\alpha$, and $\beta$ are positive constants.  Specifically, using the definition of the fractional heat operator$(\partial_t-\Delta)^\alpha$ given in \cite{T}, we obtain, when they exist,optimal pointwise upper bounds on $\mathbb{R}^n \times(0,\infty)$ fornonnegative solutions $u$ and $v$ of this initial value problem with particularemphasis on these bounds as $t\to0^+$ and as $t\to\infty$.

Pointwise Bounds and Blow-up for Systems of Nonlinear Fractional Parabolic Inequalities