In this paper, we study the self-similar solutions of the degenerate diffusion equation ut − div (∣∇um∣p−2∇um) = 0 of polytropic filtration diffusion in ℝN × (0, ±∞) or (ℝN{0}) × (0, ±∞) with N ≥ 1, m > 0, p > 1, such that m(p − 1) > 1. We give a clear classification of the self-similar solutions of the form <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_219_Article_IEq1.gif" Format="GIF" Height="26" Rendition="HTML" Resolution="72" Type="Linedraw" Width="207" /> </InlineMediaObject> <EquationSource Format="TEX">$u(x,t)=(\beta t)^{-\frac{\alpha}{\beta}}w((\beta t)^{-\frac{1}{\beta}}|x|)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow> <mo>−</mo> <mfrac> <mi>α</mi> <mi>β</mi> </mfrac> </mrow> </msup> <mi>w</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mi>β</mi> </mfrac> </mrow> </msup> <mrow> <mo stretchy="false">∣</mo> </mrow> <mi>x</mi> <mrow> <mo stretchy="false">∣</mo> </mrow> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> with α ∈ ℝ and β = α [m (p − 1) − 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.

Classification of self-similar solutions of the degenerate polytropic filtration equations

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2025年2月6日

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Zhipeng Liu,
Shanming Ji

摘要

In this paper, we study the self-similar solutions of the degenerate diffusion equation u_t − div (∣∇u^m∣^p−2∇u^m) = 0 of polytropic filtration diffusion in ℝ^N × (0, ±∞) or (ℝ^N{0}) × (0, ±∞) with N ≥ 1, m > 0, p > 1, such that m(p − 1) > 1. We give a clear classification of the self-similar solutions of the form $u(x,t)=(\beta t)^{-\frac{\alpha}{\beta}}w((\beta t)^{-\frac{1}{\beta}}|x|)$ $u (x, t) = (β t)^{- \frac{α}{β}} w ((β t)^{- \frac{1}{β}} ∣ x ∣)$ with α ∈ ℝ and β = α [m (p − 1) − 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.