Blow-up vs. Global Finiteness for an Evolution $p$-Laplace System with Nonlinear Boundary Conditions
Year: 2009
Author: Xuesong Wu, Wenjie Gao
Communications in Mathematical Research , Vol. 25 (2009), Iss. 4 : pp. 309–317
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 $(u_0, v_0)$.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2009-CMR-19348
Communications in Mathematical Research , Vol. 25 (2009), Iss. 4 : pp. 309–317
Published online: 2009-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 9
Keywords: nonlinear boundary value problem evolution p-Laplace system blow-up.