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Existence of the global solution for the parabolic Monge-Ampère equations on compact Riemannian manifolds. (English) Zbl 1237.58027
Summary: On a compact Riemannian manifold $$(M,g)$$, we consider the existence and nonexistence of global solutions for the parabolic Monge-Ampère equation $\begin{cases}\frac{\partial}{\partial t}\varphi=\log\left(\frac{\text{det}(g+\text{Hess}\,\varphi)}{\text{det}\,g}\right)-\lambda\varphi^p-f(x),\\ \varphi(x,0)=\varphi_0(x).\end{cases}\tag{*}$ Here $$p>1$$ and $$\lambda$$ are real parameters. $$-f,\varphi_0:M\to(0,+\infty)$$ are smooth functions on $$M$$. If $$\lambda>0$$, then the solution $$\varphi$$ of (*) exists for all times $$t$$ and $$\varphi_t=\varphi(\cdot ,t)$$ converges exponentially towards a solution $$\varphi_\infty$$ of its stationary equation as $$t\to\infty$$. In the case of $$\lambda<0$$, global solution of (*) does not exist. Thus we obtain the nonexistence of the positive solution for the stationary equation of (*).

##### MSC:
 58J35 Heat and other parabolic equation methods for PDEs on manifolds 35K96 Parabolic Monge-Ampère equations
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