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New blow‐up conditions to p ‐Laplace type nonlinear parabolic equations under nonlinear boundary conditions
Author(s) -
Chung SoonYeong,
Hwang Jaeho
Publication year - 2021
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.7172
Subject(s) - mathematics , eigenvalues and eigenvectors , nonlinear system , laplace transform , mathematical analysis , type (biology) , boundary value problem , laplace operator , operator (biology) , robin boundary condition , laplace's equation , boundary (topology) , neumann boundary condition , physics , biochemistry , chemistry , repressor , gene , transcription factor , ecology , quantum mechanics , biology
In this paper, we study blow‐up phenomena of the following p ‐Laplace type nonlinear parabolic equationsu t = ∇ · ρ ( | ∇ u | p ) | ∇ u | p − 2 ∇ u + f ( x , t , u ) ,in Ω × ( 0 , t ∗ ) , under nonlinear mixed boundary conditionsρ ( | ∇ u | p ) | ∇ u | p − 2∂ u ∂ n + θ ( z ) ρ ( | u | p ) | u | p − 2 u = h ( z , t , u ) ,onΓ 1 × ( 0 , t ∗ ) , and u = 0 on Γ 2 × (0, t ∗ ) such thatΓ 1 ∪ Γ 2 = ∂ Ω , where f and h are real‐valued C 1 ‐functions. To discuss blow‐up solutions, we introduce new conditions: For each x ∈ Ω , z ∈ ∂ Ω , t > 0 , u > 0 , and v > 0 ,( D p1 ) : α F ( x , t , u ) ≤ u f ( x , t , u ) + β 1u p + γ 1 ,α H ( z , t , u ) ≤ u h ( z , t , u ) + β 2u p + γ 2 ,( D p2 ) : δ v ρ ( v ) ≤ P ( v ) ,for some constants α , β 1 , β 2 , γ 1 , γ 2 , and δ satisfyingα > 2 ,δ > 0 ,β 1 +λ R + 1λ Sβ 2 ≤α δ p − 1ρ mλ R ,and 0 ≤ β 2 ≤α δ p − 1ρ mλ S , whereρ m : = inf w > 0 ρ ( w ) , P ( v ) = ∫ 0 v ρ ( w ) d w , F ( x , t , u ) = ∫ 0 u f ( x , t , w ) d w , and H ( x , t , u ) = ∫ 0 u h ( x , t , w ) d w . Here, λ R is the first Robin eigenvalue and λ S is the first Steklov eigenvalue for the p ‐Laplace operator, respectively.