Existence of the global solution for the parabolic Monge–Ampère equations on compact Riemannian manifolds

On a compact Riemannian manifold ( M , g ) , we consider the existence and nonexistence of global solutions for the parabolic Monge–Ampère equation(⁎) { ∂ ∂ t φ = log ( det ( g + Hess φ ) det g ) − λ φ p − f ( x ) , φ ( x , 0 ) = φ 0 ( x ) . Here p > 1 and λ are real parameters. − f , φ 0 : M → ( 0 , + ∞ ) are smooth functions on M. If λ > 0 , then the solution φ of (⁎) exists for all times t and φ t = φ ( ⋅ , t ) converges exponentially towards a solution φ ∞ of its stationary equation as t → ∞ . In the case of λ < 0 , it does not have the global solution of (⁎). Thus we obtain the nonexistence of the positive solution for the stationary equation of (⁎).