2014/01/28 by Huyuan Chen, Chen, Huyuan, Veron, Laurent +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1401.7187
openalex publication_date 2014/01/28 · openalex created_date 2022/02/24 · openalex updated_date 2026/07/28
We study existence and uniqueness of weak solutions to (F) ∂_t u+ (-Δ)^\alphau+h(t, u)=0 in (0,∞)×\RN,with initial condition u(0,⋅)=ν in \RN, where N≥2, the operator (-Δ)αis the fractional Laplacian with α∈(0,1), ν isa bounded Radon measure and h:(0,∞)×\R→\R is a continuous function satisfying a subcritical integrability condition.In particular, if h(t,u)=tβup with β\textgreater-1 and 0 \textless p \textless p^*_β:=1+(2α(1+β))/(N), we prove that there exists a unique weak solution u_k to (F) with ν=kδ_0, where δ_0 is the Dirac mass at the origin. We obtain that u_k→∞ in (0,∞)×\RN as k→∞ for p∈(0,1] and the limit of u_k exists as k→∞ when 1 \textless p \textless p^*_β, we denote it by u_∞.When 1+(2α(1+β))/(N+2α):=p**_β\textless p \textless p^*_β,u_∞ is the minimal self-similar solution of (F)_∞ ∂_t u+ (-Δ)αu+tβup=0 in (0,∞)×\RN with the initial condition u(0,⋅)=0 in \RN∖\0\ and it satisfies u_∞(0,x)=0 for x≠ 0.While if 1\textless p \textless p**_β, then u_∞≡ U_p, where U_p is the maximal solution of the differential equation y'+tβyp=0 on \R_+.