Remarks on the zero set of a solution of a parabolic equation in space dimension 1 (Q1127017)

Let \(u\) be a nontrivial solution of the parabolic equation \(u_t= u_{xx}+ q(x,t)u\), \((x,t)\in (0,1)\times (0,T)\), satisfying a certain mixed boundary condition. The author shows that for any point \((x_0,t_0)\in (0,1)\times (0,T)\) where \(u(x_0, t_0)= u_x(x_0, t_0)= 0\), there exists a neighbourhood in which the zero set of \(u\) only consists in a finite union of curves.