Journal of the
Korean Mathematical Society JKMS

The quaternionic Calabi conjecture was introduced by Alesker-Verbitsky, analogous to the K\"ahler case which was raised by Calabi. On a compact connected hypercomplex manifold, when there exists a  flat hyperK\"ahler metric which is compatible with the underlying hypercomplex structure, we will consider the parabolic quaternionic Monge-Amp\`{e}re equation.  Our goal is to prove the long time existence and $C^{\infty}$ convergence for normalized solutions  as $t\rightarrow\infty$. As a consequence, we show that the limit function is exactly the solution of quaternionic Monge-Amp\`{e}re equation, this gives a parabolic proof for the quaternionic Calabi conjecture in this special setting.