Interactions among multi-nonlinearities in a nonlinear diffusion system with absorptions and nonlinear boundary flux
type: Item
instance of: scholarly article
title: Interactions among multi-nonlinearities in a nonlinear diffusion system with absorptions and nonlinear boundary flux
zbMATH Open document ID: 1063.35073
author: Sining Zheng
author: Xianfa Song
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2004-09-23T00:00:00Z
review text: The following nonlinear diffusion equations with nonlinear absorption and coupled nonlinear boundary flux are studied: \[ \begin{alignedat}{3} u_t &= \Delta e^{mu} - a_1e^{\alpha_1u}, &\quad v_t &= \Delta e^{nu} - a_2e^{\beta_1v},&&\quad (x,t)\in\Omega\times (0,T),\\ \frac{\partial u}{\partial \eta} &= e^{pv + \alpha_2u}, &\quad \frac{\partial v}{\partial\eta} &= e^{qu + \beta_2v}, &&\quad (x,t)\in\partial \Omega\times (0,T),\\ u(x,0) &= u_0(x),&\quad v(x,0) &= v_0(x), &&\quad x\in\overline\Omega, \end{alignedat} \] where \(\Omega\subset\mathbb R^N\) is a bounded domain with smooth boundary \(\partial\Omega\); constants \(m\), \(n\), \(p\), \(q\), \(a_i > 0\), \(\alpha_i\), \(\beta_i\geq 0\), \((i = 1,2)\); \(u_0\) and \(v_0\) are nonnegative functions satisfying the compatibility conditions on \(\overline\Omega\). There are three kinds of nonlinear mechanisms which influence on the dynamical processes in the problem, i.e., nonlinear diffusion, nonlinear absorption, and nonlinear boundary flux. The aim of the article is to study the interactions among them. Two parameters are used to get a simple description for the interactions among the multi-nonlinearities in the system under consideration. The existence of blow up phenomenon is determined by the signs of certain parameters and the conditions for global boundedness of solutions to the above-mentioned problem are given in the terms of signs of parameters. The case of critical signs of parameters which are used to describe the behavior of the solutions is also discussed.
reviewed by: Vladimir N. Grebenev
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B33
zbMATH DE Number: 2102520
zbMATH Keywords: blow up
zbMATH Keywords: global boundedness
MaRDI profile type: Publication
full work available at URL: j.na.2004.02.026
OpenAlex ID: W1995617794
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DOI: 10.1016/J.NA.2004.02.026
description: scientific article; zbMATH DE number 2102520