DOI: 10.1515/anona-2025-0185 ISSN: 2191-950X
The method of moving planes for nonlocal uniformly fractional parabolic equations in a half space
Yanjuan Tang Abstract
In this paper, we consider the uniformly fractional parabolic equations in a half space
∂
u
∂
t
(
x
,
t
)
+
D
a
s
u
(
x
,
t
)
=
f
(
u
(
x
,
t
)
)
,
(
x
,
t
)
∈
R
+
n
×
R
,
$$\frac{\partial u}{\partial t}\left(x,t\right)+{\mathcal{D}}_{a}^{s}u\left(x,t\right)=f\left(u\left(x,t\right)\right), \left(x,t\right)\in {\mathbb{R}}_{+}^{n}{\times}\mathbb{R},$$
for
n
≥ 3. We mainly establish the monotonicity of solutions for the equations. Firstly, we derive the narrow region principle and maximum principle for antisymmetric functions under the assumptions that
u
is uniformly bounded and satisfies a growth condition, which weakens the conventional decay hypothesis
u
→ 0 at infinity. In order to investigate the monotonicity of the solutions, we implement the method of moving planes.