Blow-up phenomena for p-Laplacian parabolic equations under nonlocal boundary conditions. Issue 16 (10th December 2021) - Kou, Wei; Ding, Juntang
Abstract : The purpose of this paper is to deal with the blow-up problems of the following p -Laplacian parabolic equations with nonlocal boundary conditions: 1 h ( u ) t = ∇ ⋅ | ∇ u | p − 2 ∇ u + k 1 ( t ) f ( u ) in Ω × ( 0, t ∗ ), | ∇ u | p − 2 ∂ u ∂ ν = k 2 ( t ) ∫ Ω g ( u ) d x on ∂ Ω × ( 0, t ∗ ), u ( x, 0 ) = u 0 ( x ) ≥ 0 in Ω ¯, where p >2, Ω ⊂ R n ( n ≥ 2 ) is a bounded convex region, and the boundary ∂ Ω is smooth. With the help of differential inequality techniques and Sobolev inequalities, we prove that the blow-up does occur on some certain conditions of the data. In addition, we obtain upper bounds and lower bounds of the blow-up time in Ω ⊂ R n ( n ≥ 2 ) .