New blow‐up conditions to p ‐Laplace type nonlinear parabolic equations under nonlinear boundary conditions

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.