2018/07/26 by Audrito, Alessandro, Terracini, Susanna · 1 citation
#35B40 #35B44 #35B53 #35K57 #35R11 #58J35 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1807.10135
We investigate the local properties, including the nodal set and the nodal properties of solutions to the following parabolic problem of Muckenhoupt-Neumann type: \begincases ∂t u - y-a ∇ ⋅(ya ∇ u) = 0 amp; in \mathbbB1+ × (-1,0)
-∂ya u = q(x,t)u amp; on B1 × \0\ × (-1,0), \endcases where a∈(-1,1), is a fixed parameter \mathbbB1+⊂ 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 Hsu(x,t) = (1)/(|Γ(-s)|) ∫-∞t ∫ℝN [u(x,t) - u(z,τ)] \fracGN(x-z,t-τ)(t-τ)1+s dzdτ. 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.