Global existence and blowup of solutions of the heat equation with a memory boundary of polynomial type

Abstract

In this paper, we use the constructive methods to show that every Schwartz function can be decomposed as a product of two Schwartz functions. The same is true for D(R n) function with the same compact set. As their applications, it immediately follows that S (R n) = S (R n)S (R n) and D(R n) = D(R n)D(R n). It should be pointed out that we use the adaptive idea and method to give the decompositions.