Monotonicity of solutions for fractional parabolic equations

Abstract

In this paper, we consider the following fractional parabolic equation with a gradient nonlinear term on the upper half space \begin{equation*} \begin{cases} \frac{\partial u}{\partial t}(x, t)+(-\Delta)^s u(x, t)=f(t,x,u(x,t),\triangledown u(x,t)),& (x, t) \in \mathbb{R}_+^{n} \times \mathbb{R}, \\u(x, t) = 0,& (x, t) \notin \mathbb{R}_+^{n} \times \mathbb{R}. \end{cases} \end{equation*} Without assuming any asymptotic decay of $u$ near infinity, we prove that~$u(x,t)$~is strictly increasing with respect to $x_1$ by the method of moving planes. Meanwhile, we show the existence of positive solutions related to the fractional parabolic equations on the whole space $\mathbb{R}^{n-1}.$