A semilinear parabolic problem with singular term at the boundary

• Published in: Journal of evolution equations 2016-03, Vol.16 (1), p.131-153
• Main Authors: Abdellaoui, B., Biroud, K., Primo, A.
• Format: Article
• Language: English
• Subjects: Analysis; Mathematics; Mathematics and Statistics
• ISSN: 1424-3199; 1424-3202
• DOI: 10.1007/s00028-015-0295-1

In this paper, we deal with a class of semilinear parabolic problems related to a Hardy inequality with singular weight at the boundary. More precisely, we consider the problem P u t - Δ u = λ u p d 2 in Ω T ≡ Ω × ( 0 , T ) , u > 0 in Ω T , u ( x , 0 ) = u 0 ( x ) > 0 in Ω , u = 0 on ∂ Ω × ( 0 , T ) , where Ω is a bounded regular domain of R N , d ( x ) = dist ( x , ∂ Ω ) , p > 0 , and λ > 0 is a positive constant. We prove that If 0 < p < 1 , then (P) has no positive very weak solution. If p = 1 , then (P) has a positive very weak solution under additional hypotheses on λ and u 0 . If p > 1 , then, for all λ > 0 , the problem (P) has a positive very weak solution under suitable hypothesis on u 0 . Moreover, we consider also the concave–convex-related case.