A parabolic Monge-Amp\`ere type equation of Gauduchon metrics

We prove the long time existence and uniqueness of solution to a parabolic Monge-Amp\`ere type equation on compact Hermitian manifolds. We also show that the normalization of the solution converges to a smooth function in the smooth topology as $t$ approaches infinity which, up to scaling, is the solution to a Monge-Amp\`ere type equation. This gives a parabolic proof of the Gauduchon conjecture based on the solution of Sz\'ekelyhidi, Tosatti and Weinkove to this conjecture.