Existence and uniqueness of the singular self-similar solutions of the fast diffusion equation and logarithmic diffusion equation

2023/08/20 by Kin Ming Hui, Hui, Kin Ming
Mathematics · Medicine · #2020 Mathematics Subject Classification: Primary 35J75 #35K65 Secondary 35J70 #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2308.10221

openalex publication_date 2023/08/20 · openalex created_date 2023/08/23 · openalex updated_date 2026/07/28

Abstract

Let n≥ 3, 00, η>0, β>(mρ₁)/(n-2-nm), α=α_m=(2β+ρ₁)/(1-m), β₀>0 and α₀=2β₀+1. We use fixed point argument to give a new proof for the existence and uniqueness of radially symmetric singular solution f=f^(m) of the elliptic equation Δ(f^m/m)+αf+βx⋅∇ f=0, f>0, in ℝⁿ∖\0\, satisfying lim_{|x|→ 0}|x|^α/βf(x)=η. We also prove the existence and uniqueness of radially symmetric singular solution g of the equation Δlog g+α₀ g+β₀x⋅∇ g=0, g>0, in ℝⁿ∖\0\, satisfying lim_{|x|→ 0}|x|^α₀/β₀g(x)=η. Such equations arises from the study of backward singular self-similar solution of the fast diffusion equation u_t=Δu^m and the logarithmic diffusion equation u_t=Δlog u respectively. We will also prove the asymptotic decay rate of the function f as |x|→∞.

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• Non-existence of radially symmetric singular self-similar solutions of the fast diffusion equation
• Existence of new self-similar solutions of the fast diffusion equation
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• Asymptotic large time behavior of singular solutions of the fast diffusion equation
• Solutions with snaking singularities for the fast diffusion equation