Blow-up analysis for parabolic p-Laplacian equations with a gradient source term

Abstract

In this work, we deal with the blow-up solutions of the following parabolic p-Laplacian equations with a gradient source term: ⎧ ⎪ ⎨ ⎪ ⎩ (b(u)) t = ∇ • (|∇u| p-2 ∇u) + f (x, u, |∇u| 2 , t) in Ω × (0, t *), ∂u ∂n = 0 on ∂Ω × (0, t *), u(x, 0) = u 0 (x) ≥ 0 i n Ω, where p > 2, the spatial domain Ω ⊂ R N (N ≥ 2) is bounded, and the boundary ∂Ω is smooth. Our research relies on the creation of some suitable auxiliary functions and the use of the differential inequality techniques and parabolic maximum principles. We give sufficient conditions to ensure that the solution blows up at a finite time t *. The upper bounds of the blow-up time t * and the upper estimates of the blow-up rate are also obtained.

Blow-up results of the positive solution for a weakly coupled quasilinear parabolic system

“…In (2), Ω is a spatial bounded convex region in R n (n ≥ 2) and the boundary ∂Ω of the spatial region Ω is smooth. They mainly relied on the differential inequality technique to give some conditions to fully guarantee that the classical nonnegative solution (u, v) of ( 2) must be a blow-up solution with a finite blow-up time t * .…”