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: {partial derivative(t)u - y(-alpha)V center dot (y(alpha)Vu) = 0 in B-1(+) x (-1, 0) - partial derivative (alpha)(y) u = q (x, t ) u on B1 x {0} x (-1, 0), where (alpha) is an element of (-1, 1) is a fixed parameter,B-1(+) C R-N (+1) is the upper unit half ball and B 1 is the unit ball in 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/|Gamma(-s)| integral(t)(-infinity) integral(N)(R) [u(x, t)-u(z,tau)] G(N) (x-z, t-tau)/(t-tau)(1+s) dzd tau 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 co dimension one in R-N x 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-Po on 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.