On the Nodal Set of Solutions to a Class of Nonlocal Parabolic Equations
Authors: Audrito Alessandro , Terracini Susanna 
Published: September 2024

We investigate the local properties, including the nodal set and the nodal properties of solutions to the following parabolic problem of Muckenhoupt-Neumann type:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row 1st Column partial-differential Subscript t Baseline u overbar minus y Superscript negative a Baseline nabla dot left-parenthesis y Superscript a Baseline nabla u overbar right-parenthesis equals 0 2nd Column a m p semicolon in double-struck upper B 1 Superscript plus Baseline times left-parenthesis negative 1 comma 0 right-parenthesis 2nd Row 1st Column minus partial-differential Subscript y Superscript a Baseline u overbar equals q left-parenthesis x comma t right-parenthesis u 2nd Column a m p semicolon on upper B 1 times StartSet 0 EndSet times left-parenthesis negative 1 comma 0 right-parenthesis comma EndLayout">

{

∂
t

u
¯

−

y

−
a

∇
⋅
(

y
a

∇

u
¯

)
=
0

a
m
p
;
 in 

B

1
+

×
(
−
1
,
0
)

−

∂
y
a

u
¯

=
q
(
x
,
t
)
u

a
m
p
;
 on 

B
1

×
{
0
}
×
(
−
1
,
0
)
,

\begin{equation*} \begin {cases} \partial _t \overline {u} - y^{-a} \nabla \cdot (y^a \nabla \overline {u}) = 0 \quad &\text { in } \mathbb {B}_1^+ \times (-1,0) \\ -\partial _y^a \overline {u} = q(x,t)u \quad &\text { on } B_1 \times \{0\} \times (-1,0), \end{cases} \end{equation*}

where

a
∈
(
−
1
,
1
)

a\in (-1,1)

is a fixed parameter,

B

1
+

⊂

R

N
+
1

\mathbb {B}_1^+\subset \mathbb {R}^{N+1}

is the upper unit half ball and

B
1

B_1

is the unit ball in

R

N

\mathbb {R}^N

. Our main motivation comes from its relation with a class of nonlocal parabolic equations involving the fractional power of the heat operator
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript s Baseline u left-parenthesis x comma t right-parenthesis equals StartFraction 1 Over StartAbsoluteValue normal upper Gamma left-parenthesis negative s right-parenthesis EndAbsoluteValue EndFraction integral Subscript negative normal infinity Superscript t Baseline integral Underscript double-struck upper R Superscript upper N Baseline Endscripts left-bracket u left-parenthesis x comma t right-parenthesis minus u left-parenthesis z comma tau right-parenthesis right-bracket StartFraction upper G Subscript upper N Baseline left-parenthesis x minus z comma t minus tau right-parenthesis Over left-parenthesis t minus tau right-parenthesis Superscript 1 plus s Baseline EndFraction d z d tau period">

H
s

u
(
x
,
t
)
=

1

|

Γ
(
−
s
)

|

∫

−
∞

t

∫

R

N

[
u
(
x
,
t
)
−
u
(
z
,
τ
)
]

G
N

(
x
−
z
,
t
−
τ
)

(
t
−
τ

)

1
+
s

d
z
d
τ
.

\begin{equation*} H^su(x,t) = \frac {1}{|\Gamma (-s)|} \int _{-\infty }^t \int _{\mathbb {R}^N} \left [u(x,t) - u(z,\tau )\right ] \frac {G_N(x-z,t-\tau )}{(t-\tau )^{1+s}} dzd\tau . \end{equation*}

We characterise the possible blow-ups and we examine the structure of the nodal set of solutions vanishing with a finite order. More precisely, we prove that the nodal set has at least parabolic Hausdorff codimension one in

R

N

×

R

\mathbb {R}^N\times \mathbb {R}

, and can be written as the union of a locally smooth part and a singular part, which turns out to possess remarkable stratification properties. Moreover, the asymptotic behaviour of general solutions near their nodal points is classified in terms of a class of explicit polynomials of Hermite and Laguerre type, obtained as eigenfunctions to an Ornstein-Uhlenbeck type operator. Our main results are obtained through a fine blow-up analysis which relies on the monotonicity of an Almgren-Poon type quotient and some new Liouville type results for parabolic equations, combined with more classical results including Federer’s reduction principle and the parabolic Whitney’s extension.