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

Abstract

In this paper, we study blow‐up phenomena of the following p‐Laplace type nonlinear parabolic equations under nonlinear mixed boundary conditions and on Γ2 × (0, t∗) such that , 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, for some constants α, β1, β2, γ1, γ2, and δ satisfying where , , , and . Here, λR is the first Robin eigenvalue and λS is the first Steklov eigenvalue for the p‐Laplace operator, respectively.