Blow-up vs. Global Finiteness for an Evolution $p$-Laplace System with Nonlinear Boundary Conditions

Abstract

In this paper, the authors consider the positive solutions of the system of the evolution $p$-Laplacian equations $$\begin{cases} u_t ={\rm div}(| ∇u |^{p−2} ∇u) + f(u, v), & (x, t) ∈ Ω × (0, T ), & \\ v_t = {\rm div}(| ∇v |^{p−2} ∇v) + g(u, v), & (x, t) ∈ Ω × (0, T) \end{cases}$$with nonlinear boundary conditions $$\frac{∂u}{∂η} = h(u, v), \frac{∂v}{∂η} = s(u, v),$$and the initial data $(u_0, v_0)$, where $Ω$ is a bounded domain in $\boldsymbol{R}^n$ with smooth boundary $∂Ω, p > 2$, $h(· , ·)$ and $s(· , ·)$ are positive $C^1$ functions, nondecreasing in each variable. The authors find conditions on the functions $f, g, h, s$ that prove the global existence or finite time blow-up of positive solutions for every $(u_0, v_0)$.