On non-denseness for a method of fundamental solutions with source points fixed in time for parabolic equations

Abstract

Linear combinations of fundamental solutions to the parabolic heat equation with source points fixed in time is investigated. The open problem whether these linear combinations generate a dense set in the space of square integrable functions on the lateral boundary of a space-time cylinder, is settled in the negative. Linear independence of the set of fundamental solutions is shown to hold. It is outlined at the end, for a particular example, that such linear combinations constitute a linearly independent and dense set in the space of square integrable functions on the upper top part (where time is fixed) of the boundary of this space-time cylinder.