In this paper, we study the self-similar solutions of the degenerate diffusion equation $u_t-\mathrm{div}\left({|\nabla u^m|}^{p-2} \nabla u^m\right)=0$ of polytropic filtration diffusion in $\mathbb{R}^N \times (0,\pm\infty)$ or $(\mathbb{R}^N\backslash\{0\})\times (0,\pm\infty)$ with $N\ge1$, $m>0, p>1$, such that $m(p-1)>1$. We give a clear classification of the self-similar solutions of the form $u(x,t)=(\beta t)^{-\frac{\alpha}{\beta}}w((\beta t)^{-\frac{1}{\beta}}|x|)$ with $\alpha\in\mathbb{R}$ and $\beta=\alpha \left[m\left(p-1\right)-1\right]+p$, 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.
Zhipeng LIU
,
Shanming JI
. CLASSIFICATION OF SELF-SIMILAR SOLUTIONS OF THE DEGENERATE POLYTROPIC FILTRATION EQUATIONS[J]. Acta mathematica scientia, Series B, 2025
, 45(2)
: 615
-635
.
DOI: 10.1007/s10473-025-0219-2
[1] Audrito A, Vázquez J L. The Fisher-KPP problem with doubly nonlinear diffusion. J Differential Equations, 2017, 263: 7647-7708
[2] Barenblatt G I. On self-similar motions of a compressible fluid in a porous medium. Akad Nauk SSSR Prikl Mat Meh, 1952, 16: 679-698
[3] Barna I F. Self-similar solutions of three-dimensional N avier- S tokes equation. Commun Theoretical Physics, 2011, 56: 745-750
[4] Bidaut-Véron M F. Self-similar solutions of the p- L aplace heat equation: the fast diffusion case. Pacific J Mathematics, 2006, 227: 201-270
[5] Bidaut-Véron M F. Self-similar solutions of the $p$- L aplace heat equation: the case when $p > 2$. Proc Royal Soc Edinb, 2009, 139: 1-43
[6] Bizoń P, Breitenlohner P, Maison D, Wasserman A. Self-similar solutions of the cubic wave equation. Nonlinearity, 2009, 23: 225-236
[7] Bizoń P, Maison D, Wasserman A.Self-similar solutions of semilinear wave equations with a focusing nonlinearity. Nonlinearity , 2007, 20: 2061-2074
[8] Chen X, Qi Y, Wang M. Self-similar singular solutions of a $p$- L aplacian evolution equation with absorption. J Differential Equations, 2003, 190: 1-15
[9] Han Y, Gao W. Extinction and non-extinction for a polytropic filtration equation with a nonlocal source. Applicable Analysis, 2013, 92: 636-650
[10] Jin C, Yin J. Critical exponents and non-extinction for a fast diffusive polytropic filtration equation with nonlinear boundary sources. Nonlinear Analysis, 2007, 67: 2217-2223
[11] Jin C, Yin J, Ke Y. Critical extinction and blow-up exponents for fast diffusive polytropic filtration equation with sources. Proc Edinb Math Soc, 2009, 52: 419-444
[12] Leray J. Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Mathematica, 1934, 63: 193-248
[13] Li H, Han Y, Gao W. Critical extinction exponents for polytropic filtration equations with nonlocal source and absorption. Acta Mathematica Scientia, 2015, 35B: 366-374
[14] Li Z, Mu C. Critical exponents for a fast diffusive polytropic filtration equation with nonlinear boundary flux. J Math Anal Appl, 2008, 346: 55-64
[15] Mi Y, Mu C, Li Z. Global existence and nonexistence of the non- N ewtonian polytropic filtration equations coupled with nonlinear boundary conditions. Applicable Analysis, 2010, 89: 1789-1803
[16] Mi Y, Wang X, Mu C. Blow-up set for the non-Newtonian polytropic filtration equation subjected to nonlinear Neumann boundary condition. Applicable Analysis, 2013, 92: 1332-1344
[17] Nečas J, Růžička M, Šverák V. On L eray's self-similar solutions of the N avier- S tokes equations. Acta Mathematica, 1996, 176: 283-294
[18] Tsai T. On L eray's self-similar solutions of the Navier-Stokes equations satisfying local energy estimates. Arch Ration Mech Anal, 1998, 143: 29-51
[19] Tsai T. Forward discretely self-similar solutions of the N avier-Stokes equations. Commun Math Phys, 2014, 328: 29-44
[20] Wang J, Gao W, Su M.Periodic solutions of non- N ewtonian polytropic filtration equations with nonlinear sources. Appl Math Comp, 2010, 216: 1996-2009
[21] Wang Y, Yin J. Critical extinction exponents for a polytropic filtration equation with absorption and source. Math Meth Appl Sci, 2013, 36: 1591-1597
[22] Wang Z, Yin J, Wang C. Critical exponents of the non- N ewtonian polytropic filtration equation with nonlinear boundary condition. Applied Mathematics Letters, 2007, 20: 142-147
[23] Xu T, Ji S, Mei M, Yin J. Convergence to sharp traveling waves of solutions for Burgers-Fisher-KPP equations with degenerate diffusion. J Nonlinear Sci, 2024, 34: Art 44
[24] Ye H, Yin J. Uniqueness of self-similar very singular solution for non-newtonian polytropic filtration equations with gradient absorption. Electr J Differential Equations, 2015, 83: 1-9
[25] Yin J, Li J, Jin C. Non-extinction and critical exponent for a polytropic filtration equation. Nonlinear Analysis, 2009, 71: 347-357
[26] Zheng P, Mu C. Extinction and decay estimates of solutions for a polytropic filtration equation with the nonlocal source and interior absorption. Math Meth Appl Sci, 2013, 36: 730-743
[27] Zheng P, Mu C, Ahmed I. Cauchy problem for the non- N ewtonian polytropic filtration equation with a localized reaction. Applicable Analysis, 2014, 94: 153-168
[28] Zhou J, Mu C. Critical blow-up and extinction exponents for non- N ewton polytropic filtration equation with source. Bull Korean Math Soc, 2009, 46: 1159-1173
