On the nodal set of solutions to a class of nonlocal parabolic equations

TL;DR

This paper studies the structure and properties of the nodal set of solutions to a class of nonlocal parabolic equations, providing a detailed classification of their blow-ups, nodal set stratification, and asymptotic behavior near nodal points.

Contribution

It introduces a novel analysis of the nodal set for nonlocal parabolic equations, including blow-up characterizations, stratification, and explicit asymptotic profiles, advancing understanding of these solutions.

Findings

Nodal set has at least parabolic Hausdorff codimension one.

Nodal set decomposes into smooth and singular parts with stratification.

Asymptotic behavior near nodal points classified by Hermite and Laguerre eigenfunctions.

Abstract

We investigate the local properties, including the nodal set and the nodal properties of solutions to the following parabolic problem of Muckenhoupt-Neumann type: \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 $a\in(-1,1)$, is a fixed parameter $\mathbb{B}_1^+\subset \mathbb{R}^{N+1}$ is the upper unit half ball and $B_1$ is the unit ball in $\mathbb{R}^N$. Our main motivation comes from its relation with a class of nonlocal parabolic equations involving the fractional power of the heat operator \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 $\mathbb{R}^N\times\mathbb{R}$, 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.