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 \[0\leq(\partial_t -Δ)^αu\leq v^λ\] \[ 0\leq (\partial_t -Δ)^βv\leq u^σ\] in ${\mathbb{R}}^n\times \mathbb{R}$, $n\ge 1$, 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 $t\to 0^{+}$ and as $t\to \infty$.

arXiv comment26 pages, 2 figures. arXiv admin note: text overlap with arXiv:1901.09964