The Nodal Sets of Solutions to Parabolic Equations

In this paper, we study the parabolic equations $\partial_t u=\partial_j\left(a^{ij}(x,t)\partial_iu\right)+b^j(x,t)\partial_ju+c(x,t)u$ in a domain of $\mathbb{R}^n$ under the condition that $a^{ij}$ are Lipschitz continuous. Consider the nodal set $Z_t=\{x: u(x,t)=0\}$ at a time $t$-slice. Simple examples show that the singular set $\mathcal{S}_t=\{x: u(x,t)=|\nabla_x u|(x,t)=0\}$ may coincide with nodal set. This makes the methods used in the study of nodal sets for elliptic equations fail, rendering the parabolic case much more complicated. The current strongest results in the literature establish the finiteness of the $(n-1)$-dimensional Hausdorff measure of $Z_t$, assuming either $n=1$ by Angenent or that the coefficients are time-independent and analytic by Lin. With general coefficients, the codimension-one estimate was obtained under some doubling assumption by Han-Lin but only for space-time nodal sets. In the first part, we prove that $\mathcal{H}^{n-1}(Z_t) < \infty$ in full generality, i.e. for any dimension, with time-dependent coefficients and with merely Lipschitz regular leading coefficients $a^{ij}$. In the second part, we study the evolutionary behavior of nodal sets. When $n=1$, it is proved by Angenent that the number of nodal points is non-increasing in time. For the $n$-dimensional case, we construct examples showing that measure monotonicity fails. In contrast, we prove dimension monotonicity, i.e., the Hausdorff dimension of the nodal set is non-increasing in time. This is the first monotonicity property for nodal sets in general dimensions. All the assumptions here are sharp.