Pointwise Bounds and Blow-up for Nonlinear Fractional Parabolic Inequalities

Abstract

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-Δ)^αu\leq u^λ\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 $t\to 0^{+}$ and as $t\to \infty$.

arXiv comment38 pages, one figure