On the asymptotic behaviour of solutions of some nonlinear elliptic and parabolic equations
type: Item
instance of: scholarly article
title: On the asymptotic behaviour of solutions of some nonlinear elliptic and parabolic equations
zbMATH Open document ID: 1087.35016
DOI: 10.1016/S0362-546X(02)00166-9
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-03-11T00:00:00Z
review text: The paper deals with the nonlinear differential equations of the form \[ \Delta u + \frac{1}{q-1} \nabla (c_i | u| ^{q-2} u) = 0, \qquad (1 <q < 4) \tag{1} \] and \[ \Delta u + \frac{1}{q-1} \nabla (c_i | u| ^{q-2} u) = k\dot u, \qquad (1 < q \leq 2), \tag{2} \] which are related with the processes of diffusion-convection. Here \(u = u(x)\) with \(x = (x_1,x_2,x_3) \in D \times [0, \infty)\) in (1) and \(u = u(x,t)\) with \(x = (x_1,x_2,x_3) \in D \times [0, \infty),\) \linebreak \(t \in (0,T)\) in (2). The plane domain \(D\) is supposed to be sufficiently smooth and \(0 < T \leq \infty\) is given. Assuming boundary conditions of Dirichlet type in (1) together with an initial condition in (2) and introducing the function \(\Phi(z)\) for (1) or \(\Phi(z,t)\) for (2) defined by means of the integral of the solution over the domain \(\{(x_1,x_2) \in D, x_3=z\}\) the author proves several estimates for these functions. As the author has remarked, some results given in [\textit{C. O. Horgan, L. E. Payne}, Arch. Ration. Mech. Anal. 122, No. 2, 122--144 (1993; Zbl 0790.31002)] are used in the paper.
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 1880008
zbMATH Keywords: spatial decay estimates
zbMATH Keywords: elliptic nonlinear equations
zbMATH Keywords: parabolic nonlinear equations
zbMATH Keywords: energy methods
zbMATH Keywords: diffusion-convection
author: Ramón Quintanilla
reviewed by: Hanna Marcinkowska
MaRDI profile type: Publication
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full work available at URL: s0362-546x(02)00166-9
OpenAlex ID: W2031217815
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description: scientific article; zbMATH DE number 1880008