Global solvability of nonlinear diffusion equations with forcing at the boundary
type: Item
instance of: scholarly article
title: Global solvability of nonlinear diffusion equations with forcing at the boundary
zbMATH Open document ID: 0884.35074
author: Jeffrey R. Anderson
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-04-02T00:00:00Z
review text: This paper deals with nonnegative solutions of the model problems \[ u^m_{xx} +{\varepsilon \over n} u^n_x=u_t\text{ for }0<x<1, t>0;\;u(x,0) =u_0(x)\text{ for }0\leq x\leq 1 \] and the boundary conditions \[ u(0,t)=0,\;u^m_x (1,t)=au^p(1,t)\text{ resp. }-u^m_x (0,t)= au^p(0,t),\;u(1,t) =0\text{ for }t>0. \] Here \(a,\varepsilon,m,n,p>0\), \(u_0\in L^\infty (0,1), u_0\geq 0\). By means of the limit solutions constructed in a previous paper [Comm. Partial Differential Equations 16, No. 1, 105-143 (1991; Zbl 0738.35033)] and of differential inequalities involving the \(L^{r+1}\) norm of solutions with \(r>m\) and \(r+1\geq \max(4n,2p)\), the author shows that, under rather general conditions on the data, solutions exist for all time. Possible generalizations to a more general class of problems are discussed in the final section of the paper.
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35K60
zbMATH DE Number: 1005693
zbMATH Keywords: reaction-diffusion-convection problems
reviewed by: Denise Huet
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5305
OpenAlex ID: W2013627553
cites work: Stability and instability for solutions of the convective porous medium equation with a nonlinear forcing at the boundary. II
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DOI: 10.1006/JMAA.1997.5305
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description: scientific article; zbMATH DE number 1005693