LAUSR

In this paper, we study blow‐up phenomena of the following p‐Laplace type nonlinear parabolic equations ut=∇·ρ(|∇u|p)|∇u|p−2∇u+f(x,t,u),inΩ×(0,t∗), under nonlinear mixed boundary conditions ρ(|∇u|p)|∇u|p−2∂u∂n+θ(z)ρ(|u|p)|u|p−2u=h(z,t,u),onΓ1×(0,t∗), and u=0 on Γ2 × (0, t∗) such that Γ1∪Γ2=∂Ω , where f and h are real‐valued C1‐functions. To discuss blow‐up solutions, we introduce new conditions: For each x ∈ Ω, z ∈ ∂Ω, t > 0, u > 0, and v > 0, (Dp1):αF(x,t,u)≤uf(x,t,u)+β1up+γ1,αH(z,t,u)≤uh(z,t,u)+β2up+γ2,(Dp2):δvρ(v)≤P(v), for some constants α, β1, β2, γ1, γ2, and δ satisfying α>2,δ>0,β1+λR+1λSβ2≤αδp−1ρmλR,and0≤β2≤αδp−1ρmλS, where ρm:=infw>0ρ(w) , P(v)=∫0vρ(w)dw , F(x,t,u)=∫0uf(x,t,w)dw , and H(x,t,u)=∫0uh(x,t,w)dw . Here, λR is the first Robin eigenvalue and λS is the first Steklov eigenvalue for the p‐Laplace operator, respectively.