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

On a compact Riemannian manifold (M,g)(M,g), we consider the existence and nonexistence of global solutions for the parabolic Monge–Ampère equationequation(⁎){∂∂tφ=log(det(g+Hessφ)detg)−λφp−f(x),φ(x,0)=φ0(x). Here p>1p>1 and λ   are real parameters. −f,φ0:M→(0,+∞)−f,φ0:M→(0,+∞) are smooth functions on M  . If λ>0λ>0, then the solution φ of (⁎) exists for all times t   and φt=φ(⋅,t)φt=φ(⋅,t) converges exponentially towards a solution φ∞φ∞ of its stationary equation as t→∞t→∞. In the case of λ<0λ<0, it does not have the global solution of (⁎). Thus we obtain the nonexistence of the positive solution for the stationary equation of (⁎).