Existence of Nonnegative Solutions of Nonlinear Fractional Parabolic Inequalities

We study the existence of nontrivial nonlocal nonnegative solutions $u(x,t)$ of the nonlinear initial value problems $$(\partial_t -\Delta)^\alpha u\geq u^\lambda \quad \text{in } \mathbb{R}^n \times\mathbb{R},\,n\geq 1$$ $$u=0 \quad\text{in } \mathbb{R}^n \times(-\infty,0)$$ and $$C_1 u^\lambda \leq(\partial_t -\Delta)^\alpha u\leq C_2 u^\lambda \quad\text{in } \mathbb{R}^n \times\mathbb{R},\,n\geq1$$ $$u=0 \quad\text{in } \mathbb{R}^n \times(-\infty,0),$$ 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.