On the Scaling Limits at Zeros of Solutions of Parabolic Equations

We are concerned with the local scaling limits near zero points of solutions of general parabolic equations. We show that at each zero point of finite vanishing order, the solution is asymptotic to a homogeneous polynomial solution of the “osculating” equation with constant coefficients. This provides useful information about the local structure of zero sets of solutions. In particular, we give upper bounds for Hausdorff dimensions of zero sets of weak solutions. Our analysis combines a homothety scaling argument with the theory of nonautonomous dynamical systems.