Multinonlinear interactions in quasi-linear reaction-diffusion equations with nonlinear boundary flux
type: Item
instance of: scholarly article
title: Multinonlinear interactions in quasi-linear reaction-diffusion equations with nonlinear boundary flux
zbMATH Open document ID: 1063.35089
DOI: 10.1016/S0895-7177(04)90002-7
author: Xianfa Song
author: Sining Zheng
published in: Mathematical and Computer Modelling
publication date: 2004-08-06T00:00:00Z
review text: This paper concerns the unique classical solutions of the following initial-boundary value problems with nonlinear boundary flux for the quasilinear reaction-diffusion equations: \[ \begin{aligned} u_t=\Delta\Phi(u)\mp f(u)\quad &\text{in }\Omega\times (0,T),\\ {\partial u\over\partial n}= g(u)\quad &\text{on }\partial\Omega\times (0,T),\\ u(x,0)= u_0(x)\quad &\text{in }\Omega.\end{aligned} \] Here \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega\), \(u_0\) is a positive function, \({\partial u\over\partial n}\) denotes the outward normal derivative of \(u\), \(\Phi'(s)> 0\), which makes the equations strictly parabolic, \(\Phi(s)\), \(f(s)\), \(g(s)\) are nonnegative, and \(\Phi''(s)\geq 0\) for \(s\geq s_0\). The authors prove the existence and nonexistence of the global solutions under some conditions on \(\Phi(s)\), \(f(s)\), \(g(s)\) for \(s\) large. In the case of equation with \(-f(u)\), it is shown that if the absorption \(-f(u)\) is strong enough such that it dominates the system, then the solutions are global \((T=\infty)\). Also, it is shown by constructing blowing up subsolutions that the solutions can blow up in finite time if the corporate role of flux and diffusion is dominant. In the case of equation with \(+f(u)\), under the hypotheses that \(f(s)\), \(g(s)\), \(\Phi'(s)\), \(\Phi'(s)+ \Phi'(s)g'(s)+ \Phi''(s)g(s)\) are positive and nondecreasing, by constructing sub- and supersolutions the authors give the necessary and sufficient condition for the existence of global solution.
reviewed by: Shigeru Sakaguchi
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 2082319
zbMATH Keywords: nonlinear diffusion
zbMATH Keywords: nonlinear absorption
zbMATH Keywords: global solution
zbMATH Keywords: blow up
zbMATH Keywords: classical solutions
MaRDI profile type: Publication
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full work available at URL: s0895-7177(04)90002-7
OpenAlex ID: W1967726048
description: scientific article; zbMATH DE number 2082319