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

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.