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: { ∂ 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 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.