Existence and nonexistence of positive solutions to a fractional parabolic problem with singular weight at the boundary

We consider the problem ( P ) u t + ( - Δ ) s u = λ u p δ 2 s ( x ) in Ω T ≡ Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) in Ω , u = 0 in ( I R N \ Ω ) × ( 0 , T ) , where Ω ⊂ I R N is a bounded regular domain (in the sense that ∂ Ω is of class C 0 , 1 ), δ ( x ) = dist ( x , ∂ Ω ) , 0 < s < 1 , p...

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Bibliographic Details
Published in:Journal of evolution equations 2021-06, Vol.21 (2), p.1227-1261
Main Authors: Abdellaoui, Boumediene, Biroud, Kheireddine, Laamri, El-Haj
Format: Article
Language:English
Subjects:
Analysis
Mathematics
Mathematics and Statistics
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Summary:We consider the problem ( P ) u t + ( - Δ ) s u = λ u p δ 2 s ( x ) in Ω T ≡ Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) in Ω , u = 0 in ( I R N \ Ω ) × ( 0 , T ) , where Ω ⊂ I R N is a bounded regular domain (in the sense that ∂ Ω is of class C 0 , 1 ), δ ( x ) = dist ( x , ∂ Ω ) , 0 < s < 1 , p > 0 , λ > 0 . The purpose of this work is twofold. First We analyze the interplay between the parameters s ,  p and λ in order to prove the existence or the nonexistence of solution to problem ( P ) in a suitable sense. This extends previous similar results obtained in the local case s = 1 . Second We will especially point out the differences between the local and nonlocal cases.
ISSN:1424-3199
1424-3202
DOI:10.1007/s00028-020-00623-9