Pointwise Bounds and Blow-up for Nonlinear Fractional Parabolic Inequalities

We investigate pointwise upper bounds for nonnegative solutions $u(x,t)$ of the nonlinear initial value problem \begin{equation}\label{0.1} 0\leq(\partial_t-\Delta)^\alpha u\leq u^\lambda \quad\text{ in }\mathbb{R}^n \times\mathbb{R},\,n\geq1, \end{equation} \begin{equation}\label{0.2} u=0\quad\text{in }\mathbb{R}^n\times(-\infty,0) \end{equation} where $\lambda$ and $\alpha$ are positive constants. To do this we first give a definition---tailored for our study of this problem---of fractional powers of the heat operator $(\partial_t-\Delta)^\alpha :Y\to X$ where $X$ and $Y$ are linear spaces whose elements are real valued functions on $\mathbb{R}^n \times\mathbb{R}$ and $0<\alpha<\alpha_0$ for some $\alpha_0$ which depends on $n$, $X$ and $Y$. We then obtain, when they exist, optimal pointwise upper bounds on $\mathbb{R}^n \times(0,\infty)$ for nonnegative solutions $u\in Y$ of this initial value problem with particular emphasis on those bounds as $t\to0^+$ and as $t\to\infty$.