TY - JOUR
T1 - Blow-up vs. Global Finiteness for an Evolution $p$-Laplace System with Nonlinear Boundary Conditions
AU - Wu , Xuesong
AU - Gao , Wenjie
JO - Communications in Mathematical Research 
VL - 4
SP - 309
EP - 317
PY - 2021
DA - 2021/07
SN - 25
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/cmr/19348.html
KW - nonlinear boundary value problem, evolution p-Laplace system, blow-up.
AB - <p style="text-align: justify;">In this paper, the authors consider the positive solutions of the system of
the evolution $p$-Laplacian equations $$\begin{cases} u_t ={\rm&nbsp;div}(| ∇u |^{p−2} ∇u) + f(u, v), &amp; (x, t) ∈ Ω × (0, T ),
&amp; \\ v_t = {\rm&nbsp;div}(| ∇v |^{p−2} ∇v) + g(u, v), &amp;&nbsp;(x, t) ∈ Ω × (0, T) \end{cases}$$with nonlinear boundary conditions $$\frac{∂u}{∂η}&nbsp;= h(u, v),
\frac{∂v}{∂η} = s(u, v),$$and the initial data $(u_0, v_0)$, where $Ω$ is a bounded domain in&nbsp;$\boldsymbol{R}^n$&nbsp;with smooth
boundary $∂Ω, p &gt; 2$, $h(· , ·)$ 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 $(u_0, v_0)$.</p>