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

【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 boundaryΩ,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.更多还原

【Abstract】 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 boundaryΩ,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.更多还原