The parabolic split-type Monge-Ampere on split tangent bundle surfaces

We introduce a parabolic analogue of the elliptic split-type Monge-Amp\`ere equation developed by Fang and the author, extending Streets' twisted Monge-Amp\`ere equation. The resulting equation is fully nonlinear and non-concave. We prove long-time existence for equations whose exponents are not too far apart and give conditions for convergence to the twisted Monge-Amp\`ere when the exponents approach each other. Applications include long-time convergence on K\"ahler backgrounds and reduction to the twisted Monge-Amp\`ere equation under curvature assumptions.