Localizing the Initial Condition for Solutions of the Cauchy Problem for the Heat Equation

The Cauchy problem for the heat equation with zero right-hand side is considered. The initial function is assumed to belong to the space of tempered distributions. The problem of determining the support of the initial function from solution values at some fixed time \(T > 0\) is studied. Necessary and sufficient conditions for the support to lie in a given convex compact set are obtained. These conditions are formulated in terms of the solution’s decay rate at infinity. A sharp constant in the exponential for the Landis–Oleinik conjecture on the nonexistence of fast decaying solutions is found.