A SINGULAR ENERGY LINE OF POTENTIAL WELL ON EVOLUTIONARY $ p$-LAPLACIAN WITH LOGARITHMIC SOURCE

doi:10.1007/s10473-025-0206-7

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (2): 363-384.doi: 10.1007/s10473-025-0206-7

A SINGULAR ENERGY LINE OF POTENTIAL WELL ON EVOLUTIONARY $ p$-LAPLACIAN WITH LOGARITHMIC SOURCE

Gege Liu, Jingxue Yin, Yong Luo^*

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

收稿日期:2023-12-14 出版日期:2025-03-25 发布日期:2025-05-08

A SINGULAR ENERGY LINE OF POTENTIAL WELL ON EVOLUTIONARY $ p$-LAPLACIAN WITH LOGARITHMIC SOURCE

Gege Liu, Jingxue Yin, Yong Luo^*

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Received:2023-12-14 Online:2025-03-25 Published:2025-05-08
Contact: *Yong Luo, E-mail: Luoy@m.scnu.edu.cn
About author:Gege Liu, E-mail: 2022010125@m.scnu.edu.cn; Jingxue Yin, E-mail: yjx@scnu.edu.cn
Supported by:
Jingxue YIN's work was supported by the NSFC (12171166).

摘要/Abstract

摘要： We consider large-time behaviors of weak solutions to the evolutionary $p$-Laplacian with logarithmic source of time-dependent coefficient. We find that the weak solutions may neither decay nor blow up, provided that the initial data $u(\cdot,t_0)$ is on the Nehari manifold $\mathscr{N}:=\big\{v\in W_0^{1,p}(\Omega): I(v,t_0)=0, \|\nabla v\|_p^p\neq0 \big\}$. This is quite different from the known results that the weak solutions may blow up as $u(\cdot, t_0)\in \mathscr{N}^{-}:=\big\{v\in W_0^{1,p}(\Omega): I(v,t_0)<0\big\}$ and weak solutions may decay as $u(\cdot, t_0)\in\mathscr{N}^{+}:=\big\{v\in W_0^{1,p}(\Omega): I(v,t_0)>0\big\}$.

关键词: logarithmic source, singular energy line, large-time behavior

Abstract: We consider large-time behaviors of weak solutions to the evolutionary $p$-Laplacian with logarithmic source of time-dependent coefficient. We find that the weak solutions may neither decay nor blow up, provided that the initial data $u(\cdot,t_0)$ is on the Nehari manifold $\mathscr{N}:=\big\{v\in W_0^{1,p}(\Omega): I(v,t_0)=0, \|\nabla v\|_p^p\neq0 \big\}$. This is quite different from the known results that the weak solutions may blow up as $u(\cdot, t_0)\in \mathscr{N}^{-}:=\big\{v\in W_0^{1,p}(\Omega): I(v,t_0)<0\big\}$ and weak solutions may decay as $u(\cdot, t_0)\in\mathscr{N}^{+}:=\big\{v\in W_0^{1,p}(\Omega): I(v,t_0)>0\big\}$.

Key words: logarithmic source, singular energy line, large-time behavior

中图分类号:

35A01

Gege Liu, Jingxue Yin, Yong Luo. A SINGULAR ENERGY LINE OF POTENTIAL WELL ON EVOLUTIONARY $ p$-LAPLACIAN WITH LOGARITHMIC SOURCE[J]. 数学物理学报(英文版), 2025, 45(2): 363-384.

Gege Liu, Jingxue Yin, Yong Luo. A SINGULAR ENERGY LINE OF POTENTIAL WELL ON EVOLUTIONARY $ p$-LAPLACIAN WITH LOGARITHMIC SOURCE[J]. Acta mathematica scientia,Series B, 2025, 45(2): 363-384.

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A SINGULAR ENERGY LINE OF POTENTIAL WELL ON EVOLUTIONARY $ p$-LAPLACIAN WITH LOGARITHMIC SOURCE

A SINGULAR ENERGY LINE OF POTENTIAL WELL ON EVOLUTIONARY $ p$-LAPLACIAN WITH LOGARITHMIC SOURCE

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