A strong unique continuation theorem for parabolic equations
The main result of the paper is the strong unique continuation for solutions in \(W^{2,1} _{p; \text{loc}}\) of the system of linear second-order inequalities \[ \Biggl|{{\partial u} \over {\partial t}} - D \Delta u\Biggr|\leq M_1 |\nabla u|+ M_0 |u| \] defined in a finite strip \(\mathbb R^n \times (T_1,T_2)\). As a consequence the author obtains upper bounds in the parabolic Hausdorff dimension for the nodal set of \(u\) and for the set where both \(u=0\) and \(\nabla u =0,\) and for the standard Hausdorff dimension of the set of \(x \in \mathbb R^n\) such that \(u(x,t)=0\) for fixed \(t \in (T_1, T_2).\) The method of proof relies on the classification (in self-similar variables) of the local asymptotics of the solution. Related results can be found in \textit{Q. Han} and \textit{F. H. Lin} [Commun. Pure Appl. Math. 47, 1219--1238 (1994; Zbl 0807.35052)], and \textit{C. C. Poon} [Commun. Partial Differ. Equations 21, 521--539 (1996; Zbl 0852.35055)].
scientific article; zbMATH DE number 1221866