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

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