Entire solutions to the parabolic Monge--Ampère equation with unbounded nonlinear growth in time

2023/05/15 by An, Ning, Bao, Jiguang, Liu, Zixiao · 1 citation
#35A01 #35B40 #35K96 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2305.08329

Abstract

The Liouville type theorem on the parabolic Monge--Ampère equation -u_tdet D²u=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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