2020/07/14 by Kin Ming Hui, Hui, Kin Ming, Jinwan Park +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2007.06830
openalex publication_date 2020/07/14 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
For n\≥ 3, 0<m<\(n-2)/(n), \β<0 and \α=\(2\β)/(1-m),\nwe prove the existence, uniqueness and asymptotics near the origin of the\nsingular eternal self-similar solutions of the fast diffusion equation in\n(\ℝn\∖ 0 )\× \ℝ of the form\nU\λ(x,t)=e-\α tf\λ(e-\β tx), x\∈\n\ℝn\∖ 0 , t\∈\ℝ, where f\λ is a radially\nsymmetric function satisfying
fracn-1m
Delta fm+
alpha f+
beta\nx
cdot
nabla f=0
text in
mathbbRn
setminus
0
, with\n underset substackr\→ 0\lim fracr2f(r)1-m\log\nr-1=\(2(n-1)(n-2-nm))/(|\β|(1-m)) and\n underset substackr\→\∞\limr\(n-2)/(m)f(r)=\λ\(2)/(1-m)-\(n-2)/(m),\nfor some constant \λ>0.\n As a consequence we prove the existence and uniqueness of solutions of Cauchy\nproblem for the fast diffusion equation ut=\(n-1)/(m)\Δ um in\n(\ℝn\∖ 0 )\× (0,\∞) with initial value u0\nsatisfying f\λ1(x)\≤ u0(x)\≤ f\λ2(x), \∀\nx\∈\ℝn\∖ 0 , which satisfies U\λ1(x,t)\≤ν(x,t)\≤ U\λ2(x,t), \∀ x\∈ \ℝn\∖ 0 , t\≥\n0, for some constants \λ1>\λ2>0.\n We also prove the asymptotic behaviour of such singular solution u of the\nfast diffusion equation as t\→\∞ when n=3,4 and \(n-2)/(n+2)\≤\nm<\(n-2)/(n) holds. Asymptotic behaviour of such singular solution u of\nthe fast diffusion equation as t\→\∞ is also obtained when 3\≤ n<8,\n1-\√(2/n)\≤ m<\min\(\(2(n-2))/(3n),\(n-2)/(n+2)\), and\nu(x,t) is radially symmetric in x\∈\ℝn\∖ 0 for any\nt>0 under appropriate conditions on the initial value u0.\n