On the Nodal Set of Solutions to a Class of Nonlocal Parabolic Equations

We investigate the local properties, including the nodal set and the nodal properties of solutions to the following parabolic problem of Muckenhoupt-Neumann type:(Formula Presented), where a ∈ (−1, 1) is a fixed parameter, B+1 ⊂ RN+1 is the upper unit half ball and B1 is the unit ball in RN. Our main motivation comes from its relation with a class of nonlocal parabolic equations involving the fractional power of the heat operator (Formula Presented) 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 RN × 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.