数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (2): 615-635.doi: 10.1007/s10473-025-0219-2

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CLASSIFICATION OF SELF-SIMILAR SOLUTIONS OF THE DEGENERATE POLYTROPIC FILTRATION EQUATIONS

Zhipeng Liu, Shanming Ji*   

  1. School of Mathematics, South China University of Technology, Guangzhou 510641, China
  • 收稿日期:2024-03-13 修回日期:2024-06-04 出版日期:2025-03-25 发布日期:2025-05-08

CLASSIFICATION OF SELF-SIMILAR SOLUTIONS OF THE DEGENERATE POLYTROPIC FILTRATION EQUATIONS

Zhipeng Liu, Shanming Ji*   

  1. School of Mathematics, South China University of Technology, Guangzhou 510641, China
  • Received:2024-03-13 Revised:2024-06-04 Online:2025-03-25 Published:2025-05-08
  • Contact: *Shanming Ji, E-mail: jism@scut.edu.cn
  • About author:Zhipeng Liu, E-mail: 202120130322@mail.scut.edu.cn
  • Supported by:
    Ji's research was supported by the NSFC (12271178, 12171166), the Guangzhou Basic and Applied Basic Research Foundation (2024A04J2022) and the TCL Young Scholar (2024-2027).

摘要: In this paper, we study the self-similar solutions of the degenerate diffusion equation utdiv(|um|p2um)=0 of polytropic filtration diffusion in RN×(0,±) or (RN{0})×(0,±) with N1, m>0,p>1, such that m(p1)>1. We give a clear classification of the self-similar solutions of the form u(x,t)=(βt)αβw((βt)1β|x|) with αR and β=α[m(p1)1]+p, regular or singular at the origin point. The existence and uniqueness of some solutions are established by the phase plane analysis method, and the asymptotic properties of the solutions near the origin and the infinity are also described. This paper extends the classical results of self-similar solutions for degenerate p-Laplace heat equation by Bidaut-Véron [Proc Royal Soc Edinburgh, 2009, 139: 1-43] to the doubly nonlinear degenerate diffusion equations.

关键词: self-similar solutions, polytropic filtration equation, degenerate diffusion equation, doubly nonlinear diffusion

Abstract: In this paper, we study the self-similar solutions of the degenerate diffusion equation utdiv(|um|p2um)=0 of polytropic filtration diffusion in RN×(0,±) or (RN{0})×(0,±) with N1, m>0,p>1, such that m(p1)>1. We give a clear classification of the self-similar solutions of the form u(x,t)=(βt)αβw((βt)1β|x|) with αR and β=α[m(p1)1]+p, regular or singular at the origin point. The existence and uniqueness of some solutions are established by the phase plane analysis method, and the asymptotic properties of the solutions near the origin and the infinity are also described. This paper extends the classical results of self-similar solutions for degenerate p-Laplace heat equation by Bidaut-Véron [Proc Royal Soc Edinburgh, 2009, 139: 1-43] to the doubly nonlinear degenerate diffusion equations.

Key words: self-similar solutions, polytropic filtration equation, degenerate diffusion equation, doubly nonlinear diffusion

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  • 35K65