Monotonicity results for fractional parabolic equations in the whole space

Linfeng Luo; Zhengce Zhang

doi:10.3934/dcdsb.2022235

Article Contents

2023, Volume 28, Issue 6: 3662-3679. Doi: 10.3934/dcdsb.2022235

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Monotonicity results for fractional parabolic equations in the whole space

Linfeng Luo^1, and
Zhengce Zhang^1, ,

1.
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China

^*Corresponding author: Zhengce Zhang
^*Corresponding author: Zhengce Zhang

Received: September 2022

Revised: November 2022

Early access: December 2022

Published: June 2023

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Abstract

In this paper, we consider the properties of solutions for the following fractional parabolic equation

$ \begin{equation*} \frac{\partial u}{\partial t}(x, t)+(-\Delta)^su(x, t) = f(u(x, t)), \ (x, t)\in\mathbb{R}^N\times\mathbb{R}. \end{equation*} $

Without assuming any decay behavior of $ u $ near infinity, we first establish a narrow region principle in unbounded domains, then employing the direct method of moving planes, we derive the monotonicity, antisymmetry and non-existence of the solutions. In most previous articles, to carry out the direct method of moving planes in the whole space, they often needed to assume that the solution tends to zero near infinity or that the solution is divided by a function to make the new solution satisfy such an asymptotic decay. Here we develop a new approach–estimating the singular integrals defining $ (-\Delta)^s $ and the derivative of the solution with respect to time along a sequence of approximate extreme points.

We believe that the new method employed here will be very helpful to study a class of parabolic equations involving nonlinear nonlocal operators.

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Mathematics Subject Classification: Primary: 35R11, 35B50, 35B08; Secondary: 35K58.

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