Blow-up solution for the degenerate quasilinear parabolic system with nonlinear boundary condition
type: Item
instance of: scholarly article
title: Blow-up solution for the degenerate quasilinear parabolic system with nonlinear boundary condition
zbMATH Open document ID: 1090.35092
author: Zhiwen Duan
author: Chunhong Xie
author: Li Zhou
publication date: 2006-02-16T00:00:00Z
review text: The authors consider the following system \[ u_t= u^{\alpha_1} u_{xx},\;v_t= v^{\alpha_2} v_{xx}\quad\text{for }x\in (0,\ell),\;t> 0,\tag{1} \] \[ -u_x(0,t)= av^p(0,t),\;-v_x(0,t)= bu^q(0,t),\;u(\ell,t)= v(\ell, t)= 0,\tag{2} \] \[ u(x,0)= u_0(x),\;v(x,0)= v_0(x)\quad\text{for }x\in (0,\ell),\tag{3} \] where \(0< \alpha_1\), \(\alpha_2< 1\), \(p,q,a,b> 0\), and the initial values \(u_0(x)\), \(v_0(x)> 0\) for \(x\in (0,\ell)\) are sufficiently smooth. Under compatibility conditions for (1)--(3) the authors prove the local existence and the uniqueness for (1)--(3). Moreover, they discuss the blow-up solution of (1)--(3) as well as the blow-up point set and the blow-up rate. The existence of the self-similar blow-up solution is discussed.
reviewed by: Messoud Efendiev
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B44
Mathematics Subject Classification ID: 34B05
zbMATH DE Number: 5006973
zbMATH Keywords: quasilinear parabolic system
zbMATH Keywords: nonlinear boundary condition
zbMATH Keywords: global solution
zbMATH Keywords: blow-up rate
zbMATH Keywords: self-similar blow-up solution
MaRDI profile type: Publication
full work available at URL: j.camwa.2005.01.027
OpenAlex ID: W2013573434
cites work: Q3221003
cites work: Q3733325
cites work: Uniform blow-up profile for a degenerate parabolic system with nonlocal source
cites work: Global and blow‐up solutions for non‐linear degenerate parabolic systems
cites work: Q4764290
cites work: The blow-up rate for a system of heat equations completely coupled in the boundary conditions
cites work: Q5562267
cites work: Stabilization of solutions of a degenerate nonlinear diffusion problem
cites work: Quasilinear parabolic systems under nonlinear boundary conditions
cites work: On Asymptotic Self-Similar Behaviour for a Quasilinear Heat Equation: Single Point Blow-Up
cites work: Q3689563
cites work: Blow-up of solutions of nonlinear degenerate parabolic equations
cites work: Nonlinear elliptic boundary-value problems in unbounded domains
cites work: Q5665539
cites work: Q5520962
cites work: Differential inequalities and maximum principles: theory, new methods and applications
cites work: On elliptic equations related to self-similar solutions for nonlinear heat equations
DOI: 10.1016/J.CAMWA.2005.01.027
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 5006973