Existence of Nonnegative Solutions of Nonlinear Fractional Parabolic Inequalities

Abstract

We study the existence of nontrivial nonlocal nonnegative solutions $u(x,t)$ of the nonlinear initial value problems \[ (\partial_t -Δ)^αu\geq u^λ\quad \text{in } \mathbb{R}^n \times\mathbb{R},\,n\geq 1 \] \[ u=0 \quad\text{in } \mathbb{R}^n \times(-\infty,0) \] and \[ C_1 u^λ\leq(\partial_t -Δ)^αu\leq C_2 u^λ\quad\text{in } \mathbb{R}^n \times\mathbb{R},\,n\geq1 \] \[ u=0 \quad\text{in } \mathbb{R}^n \times(-\infty,0), \] where $\lambda ,\alpha ,C_1$, and $C_2$ are positive constants with $C_1<C_2$. We use the definition of the fractional heat operator $({\partial }_t-\Delta {)}^{\alpha }$ given in [Taliaferro, 2020] and compare our results in the classical case $\alpha =1$ to known results.

arXiv comment25 pages, 1 figure