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

Abstract

We investigate nonnegative solutions $u(x,t)$ and $v(x,t)$ of the nonlinear system 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 $\mathbb{R}^n \times(0,\infty)$ for nonnegative solutions $u$ and $v$ of this initial value problem with particular emphasis on these bounds as $t\to0^+$ and as $t\to\infty$.