  In this work we find a solution to problem of the heat equation which is
annihiliated at a cubic boundary $f$. The solution turns out to be the
convolution between the fundamental solution of the heat equation and a
function $\phi$ which solves a third order ODE. However we believe that the
main contribution is the procedure itself, which links in a rather
straightforward way, solutions of the heat equation $v$ with moving boundaries
$f$ through the convolution of the heat kernel with $\mathbb{C}^p$ funtions
$\phi$.
