Entire solutions and asymptotic behavior to a class of parabolic $ k $-Hessian equations

Ning Cao; Feida Jiang

doi:10.3934/dcds.2025140

Article Contents

2026, Volume 48: 196-230. Doi: 10.3934/dcds.2025140

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Entire solutions and asymptotic behavior to a class of parabolic $ k $-Hessian equations

Ning Cao^1, and
Feida Jiang^{1, 2, ,}

1.
School of Mathematics and Shing-Tung Yau Center of Southeast University, Southeast University, Nanjing 211189, China
2.
Shanghai Institute for Mathematics and Interdisciplinary Sciences, Shanghai, 200433, China

^*Corresponding author: Feida Jiang
^*Corresponding author: Feida Jiang

Received: April 2025

Revised: August 2025

Early access: September 3, 2025

Published online: September 3, 2025

This work is supported by the National Natural Science Foundation of China [No. 12271093], the Jiangsu Provincial Scientific Research Center of Applied Mathematics [No. BK20233002] and the Start-up Research Fund of Southeast University [No. 4007012503].

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Abstract

In this paper, we conduct a systematic investigation of the existence, uniqueness and nonexistence of entire separable variable radial solutions to the following class of parabolic $ k $-Hessian equations with the parameter $ \alpha $:

$ -u_t[S_k(D^2u)]^\alpha = 1. $

We also study the asymptotic behavior and its refined version of the solution at infinity. We particularly note that our results fully generalize the result of An, Bao et al. in Nonlinear Anal. 239:113441 (2024) for the parabolic Monge-Ampère equation in the form of $ -u_t\det D^2u = 1 $. The main tools employed in this work comprise Euler's broken line method, local boundedness estimates, Keller-Osserman type criteria, the generalized L'Hospital's rule, and asymptotic stability analysis.

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Mathematics Subject Classification: Primary: 35A01, 35K96; Secondary: 35B40.

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Table 1. Main results classified by $ \alpha $ and $ k $ with $ \alpha^* = \max\{\frac{2k-n}{kn},0\} $.

Main results	ranges of $ \alpha $ and $ k $	Description
$\begin{array}{c} {\text {Local existence}}\\ {( \text{Lemma 3.1} )} \end{array}$	$ \alpha\neq0 $	$ \varphi\in C^2[0,l] $ for some $ l>0 $
$ \begin{array}{c}{\text{Entire existence}}\\{( \text{Theorem 1.1, Remark 4.1 )}} \end{array}$	$ \alpha\in(0, +\infty) $	$ u(x,t)=w(t)\varphi(\|x\|)\in C^\infty(\mathbb{R}^{n+1}_{-})$ $ w(t)=(1+k\alpha)^\frac{1}{1+k\alpha}(-t)^\frac{1}{1+k\alpha}$
	$ \alpha=-\frac{1}{k} $	$u(x,t)=w(t)\varphi(\|x\|)\in C^\infty(\mathbb{R}^{n+1}_{-}) $ $ w(t) = e^{-t}$
Entire nonexistence (Theorem 1.1)	$ \alpha\in (-\frac{1}{k},0) $	——
$\begin{array}{c}{\text {Uniqueness}}\\ {( \text{Theorems 1.2 and 1.3)}}\end{array}$	$ \alpha\in (0,+\infty) $	uniqueness of u up to a rescaling and transformation
$ \begin{array}{c}{\text{Asymptotic behavior}}\\{\text{(Theorems 1.2, 1.3, Remarks 1.2, 5.2)}} \end{array}$	$ \alpha\in(\alpha^*, +\infty) $	$ \varphi(\|x\|)=O \big( \|x\|^\frac{2k\alpha}{1+k\alpha} \big) $
	$ \alpha =\frac{2k-n}{kn}, k\in (\frac{n}{2}, n] $	$ \left( \frac{\varphi(\|x\|)}{\|x\|^{2-\frac{n}{k}}} \right)^\frac{2k^2}{2k-n} = O \big( \ln\varphi(\|x\|) \big) $
	$ \begin{array}{c}\alpha\in(0,\frac{2k-n}{kn}), \\ k\in (\frac{n}{2}, n]\end{array}$	$ \varphi(\|x\|) = O \big( \|x\|^{2-\frac{n}{k}} \big) $
	$ \alpha = +\infty $	$ \varphi_{\infty}(\|x\|)=O \big( \|x\|^2 \big) $
$ \begin{array}{c}\text{Refined behavior}\\{( \text{Theorems 1.4 and 1.5} )} \end{array}$	$ \alpha=1 $	$ \varphi(\|x\|) = c_{n,k}\|x\|^\frac{2k}{k+1}+O \big( \|x\|^{-M_{n,k}} \big) $
	$ \alpha=\frac{1}{k} $	$ \varphi(\|x\|) = c_{n,k}'\|x\|+O \big( \|x\|^{-M_{n,k}'} \big) $

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