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arXiv · 2305.08329

Entire solutions to the parabolic Monge--Amp\`ere equation with unbounded nonlinear growth in time

Ning An·Jiguang Bao·Zixiao Liu
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Abstract

The Liouville type theorem on the parabolic Monge--Amp\`ere equation $-u_t\det D^2u=1$ states that any entire parabolically convex classical solution must be of form $-t+|x|^2/2$ up to a re-scaling and transformation, under additional assumption that partial derivative with respect to time variable $u_t$ is strictly negative and bounded. In this paper, we study the case when $u_t$ is unbounded, prove an existence result of entire parabolically convex smooth solution and investigate the asymptotic behavior near infinity.

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Ning An, Jiguang Bao, Zixiao Liu. 2023-05-15. Entire solutions to the parabolic Monge--Amp\`ere equation with unbounded nonlinear growth in time. https://arxiv.org/abs/2305.08329

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