# zbMATH — the first resource for mathematics

A strong unique continuation theorem for parabolic equations. (English) Zbl 0990.35028
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 Q. Han and F. H. Lin [Commun. Pure Appl. Math. 47, 1219–1238 (1994; Zbl 0807.35052)], and C. C. Poon [Commun. Partial Differ. Equations 21, 521–539 (1996; Zbl 0852.35055)].

##### MSC:
 35B60 Continuation and prolongation of solutions to PDEs 35K40 Second-order parabolic systems 35R45 Partial differential inequalities and systems of partial differential inequalities 35K10 Second-order parabolic equations
Full Text: