EXISTENCE OF SOLUTIONS TO THE PARABOLIC EQUATION WITH A SINGULAR POTENTIAL OF THE SOBOLEV-HARDY TYPE  

作　　者：韩军强 王永达 钮鹏程 

机构地区：[1]Department of Applied Mathematics,Northwestern Polytechnical University

出　　处：《Acta Mathematica Scientia》2012年第5期1901-1918,共18页数学物理学报（B辑英文版）

基　　金：supported by NPU Foundation for Fundamental Research (NPU-FFR-JC201124);NSF of China (10871157,11001221,11002110);Specialized Research Fund for the Doctoral Program in Higher Education (200806990032)

摘　　要：We study the existence of solutions to the following parabolic equation{ut-△pu=λ/｜x｜s｜u｜q-2u,（x,t）∈Ω×（0,∞）,u（x,0）=f（x）,x∈Ω,u（x,t）=0,（x,t）∈Ω×（0,∞）,（P）}where-△pu ≡-div（｜▽u｜p-2▽u）,1〈p〈N,0〈s≤p,p≤q≤p＊（s） = N-s/N-pp,Ω is a bounded domain in RN such that 0∈Ω with a C1 boundaryΩ,f≥0 satisfying some convenient regularity assumptions.The analysis reveals that the existence of solutions for（P） depends on p,q,s in general,and on the relation between λ and the best constant in the Sobolev-Hardy inequality.We study the existence of solutions to the following parabolic equation{ut-△pu=λ/｜x｜s｜u｜q-2u,（x,t）∈Ω×（0,∞）,u（x,0）=f（x）,x∈Ω,u（x,t）=0,（x,t）∈Ω×（0,∞）,（P）}where-△pu ≡-div（｜▽u｜p-2▽u）,1〈p〈N,0〈s≤p,p≤q≤p＊（s） = N-s/N-pp,Ω is a bounded domain in RN such that 0∈Ω with a C1 boundaryΩ,f≥0 satisfying some convenient regularity assumptions.The analysis reveals that the existence of solutions for（P） depends on p,q,s in general,and on the relation between λ and the best constant in the Sobolev-Hardy inequality.

关 键 词：nonlinear parabolic equations EXISTENCE Sobolev-Hardy inequality singular potential 

分 类 号：O241.82[理学—计算数学]