On Approximating Properties of Solutions of the Heat Equation

摘要

By the maximal principle for the heat equation, a solution corresponding to zero initial data and produced by a positive Dirichlet boundary control is positive (i.e., belongs to the cone of positive functions). The notice is devoted to the question: Is the set of such solutions dense in the cone? The answer turns out to be negative: in 1 -D case we construct an explicit example of a positive function separated from this set by a positive L2-distance.