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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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