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 $$\begin{aligned} \left\{ \begin{array}{ll} (\partial _t -\Delta )^\alpha u\ge u^\lambda &{}\quad \mathrm{in}\,\, {\mathbb {R}}^n \times {\mathbb {R}},\,n\ge 1 \\ u=0 &{}\quad \mathrm{in}\,\, {\mathbb {R}}^n \times (-\infty ,0) \end{array}\right. \end{aligned}$$and $$\begin{aligned} \left\{ \begin{array}{ll} C_1 u^\lambda \le (\partial _t -\Delta )^\alpha u\le C_2 u^\lambda &{}\quad \mathrm{in}\,{\mathbb {R}}^n \times {\mathbb {R}},\,n\ge 1 \\ u=0 &{}\quad \mathrm{in}\, {\mathbb {R}}^n \times (-\infty ,0), \end{array}\right. \end{aligned}$$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 (J Math Pures Appl 133:287–328, 2020) and compare our results in the classical case \(\alpha =1\) to known results.