On a family of nonzero solutions to the heat equation ut = △u on Rn+1 which vanish on an arbitrary n-dimensional hyperplane P ⊂ Rn+1

Motivated by the paper Rosenbloom-Widder [A temperature function which vanishes initially, Amer. Math. Mon. 65(8) (1958) 607-609], we construct a more general family of nonzero solutions to the 1-dimensional heat equation u(t) = u(xx) on R-2 with zero initial data. These solutions are analytically more manageable than the well-known Tychonoff power series solution. Nonzero solutions to the n-dimensional heat equation u(t) = triangle(u) on Rn+1 with zero initial data (i.e. vanishing on the hyperplane P : {t = 0}) can be easily obtained as a consequence of the examples on R-2. Finally, given an arbitrary hyperplane P subset of Rn+1, we can construct a family of nonzero solutions which vanish on P subset of Rn+1.